Cubic Bridgeless Graphs and Braces

Cubic bridgeless graphs and braces Andrea Jiménez∗ Mihyun Kang† CIMFAV, Facultad de Ingenierı́a Universidad de Valparaı́so, Chile [email protected] Institut für Optimierung und Diskrete Mathematik Technische Universität Graz [email protected] Martin Loebl‡ Department of Applied Mathematics & Institute for Theoretical Computer Science Charles University [email protected] Abstract There are many long-standing open problems on cubic bridgeless graphs, for instance, Jaeger’s directed cycle double cover conjecture. On the other hand, many structural properties of braces have been recently discovered. In this work, we bijectively map the cubic bridgeless graphs to braces which we call the hexagon graphs, and explore the structure of hexagon graphs. We show that hexagon graphs are braces that can be generated from the ladder on 8 vertices using two types of McCuaig’s augmentations. In addition, we present a reformulation of Jaeger’s directed cycle double cover conjecture in the class of hexagon graphs. 1 Introduction Jaeger’s directed cycle double cover conjecture [1], usually known as DCDC conjecture, is broadly considered to be among the most important open problems in graph theory. A typical formulation asks whether every 2-connected graph admits a family of cycles such that one may prescribe an orientation on each cycle of the family in such a way that each edge e of the graph belongs to exactly two cycles and these cycles induce opposite orientations on e. In order to prove the DCDC conjecture, a wide variety of approaches have arisen [1, 10], among them, the topological approach. The topological approach claims that the DCDC conjecture is equivalent to the statement that every cubic bridgeless graph admits an embedding in a closed orientable surface such that every edge belongs to exactly two distinct face boundaries defined by the embedding; that is, with no dual loop. In this work, we formulate the DCDC conjecture as a problem of existence of special perfect matchings in a class of graphs that we call hexagon graphs. Initially, our motivation for the formulation of the DCDC conjecture on hexagons are critical embeddings [4, 8], that in particular are embeddings with no dual loop. The main goal of this work is to discuss recent progress on the study of the structure of hexagon graphs. The class of hexagon graphs of cubic bridgeless graphs turns out to be a subclass of braces. ∗ Partially supported by CONICYT: FONDECYT/POSTDOCTORADO 3150673, Núcleo Milenio Información y Coordinación en Redes ICM/FIC RC130003, Chile, FAPESP (Proc. 2013/03447-6) and CNPq (Proc. 456792/2014-7), Brazil. † Partially supported by the German Research Foundation (KA 2748/2-1 and KA 2748/3-1). ‡ Partially supported by the Czech Science Foundation under the contract number P202-13-21988S. 1 The class of braces, along with bricks, are a fundamental class of graphs in matching theory, mainly because they are building blocks of a perfect matching decomposition procedure; namely of the tight cut decomposition procedure [5]. In [7], McCuaig introduced a method for generating all braces starting from a large base set of graphs and recursively making use of 4 distinct types of operations. In this paper, we show that hexagon graphs are braces that can be generated from the ladder on 8 vertices using 2 types of McCuaig’s operations. In the following, we make precise the notions discussed above and formally state our main result. 1.1 Hexagon graphs Hexagon graphs are the main ingredient and the center of attention of this work. In this section, we define the class of hexagon graphs, look over some of its fundamental properties and formulate the DCDC conjecture as a question about this new class of graphs. Despite our original motivation for this new formulation of the DCDC conjecture are critical embeddings, in this work we do not introduce this notion, and we present the details and proofs regarding the formulation using rotation systems of graphs, a well known and convenient combinatorial representation of embeddings on closed orientable surfaces [9, §3.2]. The advantage of using rotation systems is that we avoid topological arguments and present the equivalence to the DCDC conjecture in a purely combinatorial way. We refer to the complete bipartite graph K3,3 as a hexagon and say that a bipartite graph H has a hexagon h if h is a subgraph of H. For a graph G and a vertex v of G, let NG (v) denote the set of neighbors of v in G. Definition 1 (Hexagon Graphs). Let G be a cubic graph with vertex set V and edge set E. A hexagon graph of G is a graph H obtained from G following the next rules: 1. We replace each vertex v in V by a hexagon hv of H so that for every pair u, v ∈ V , if u 6= v, then hu and hv are vertex disjoint. Moreover, V (H) = {V (hv ) : v ∈ V }. 2. For each vertex v ∈ V, let {vi : i ∈ Z6 } denote the vertex set of hv and {vi vi+1 , vi vi+3 : i ∈ Z6 } its edge set. With each neighbor u of v in G, we associate an index iv(u) from the set {0, 1, 2} ⊂ Z6 so that if NG (v) = {u, w, z}, then iv(u) , iv(w) , iv(z) are pairwise distinct. 3. See Figure 1. Let X = ∪v∈V {v2i : i ∈ Z6 } and Y = ∪v∈V {v2i+1 : i ∈ Z6 }. We replace each edge uv in E by two vertex disjoint edges euv , e′uv so that if both viv(u) , uiu(v) belong to either X or Y , then euv = viv(u) uiu(v) +3 , e′uv = viv(u) +3 uiu(v) . Otherwise, euv = viv(u) uiu(v) , e′uv = viv(u) +3 uiu(v) +3 . Moreover, E(H) = {E(hv ) : v ∈ V } ∪ {euv , e′uv : uv ∈ E}. We say that hv is the hexagon of H associated with the vertex v of G and that {hv : v ∈ V } is the set of hexagons of H. For uv ∈ E, we say that hu and hv are hexagon-neighbors in H. We shall refer to the S set of edges v∈V {vi vi+3 : i ∈ Z6 } as the set of red edges of H, to the set of edges {euv , e′uv : uv ∈ E} as S the set of white edges of H and finally, to the set of edges v∈V {vi vi+1 : i ∈ Z6 } as the set of blue edges of H (see Figure 1). Moreover, we shall say that a perfect matching of H containing only blue edges is a blue perfect matching. Observation 1. Hexagon graphs of cubic graphs are bipartite. Proof. Let H be a hexagon graph of a cubic bridgeless graph. Let X, Y be the sets defined in Definition 1, item 3. Note that {X, Y } is a partition of V (H) and that there are no edges connecting vertices of the same partition class. 2 u2 u1 v0 v1 u4 u0 w2 ewv euv u3 w4 w1 v5 v2 w0 u5 v3 z0 e′uv w3 v4 z1 e′wv e′zv ezv z5 w5 z2 z3 z4 Figure 1: Local representation of the hexagon-neighborhood of a hexagon hv in a hexagon graph H of a cubic graph G. The hexagon hv is associated with vertex v, where NG (v) = {u, w, z}. Red edges are depicted as red lines, blue edges are depicted as blue lines and white edges as black lines. The set X is represented by filled-in white vertices and the set Y by filled-in black vertices. Moreover, iv(u) = 0, iv(w) = 1, iv(z) = 2, iu(v) = 0, iw(v) = 2 and iz(v) = 2. The following observation is straightforward. Observation 2. Let G be a cubic graph and H be a hexagon graph of G. The following properties hold. 1. H is a 4-regular graph. 2. No white edge of H connects two vertices of the same blue hexagon. 3. Both, the set of red edges of H and the set of white edges of H form a perfect matching of H. 4. Let |V (G)| denote the cardinality of V (G). There are 2|V (G)| distinct blue perfect matchings. In the next statement we see that for each cubic bridgeless graph G, there exists a unique (up to isomorphism) hexagon graph. Note that the existence of an hexagon graph for each cubic graph is trivial since the choice of the indices iv(u) , iv(w) , iv(z) in Definition 1 for each vertex v is pairwise independent. In other words, there is an injective map from the set of all cubic bridgeless graphs to a special subset of braces (see also Theorem 8). Proposition 1. Let H and H ′ be hexagon graphs of the cubic graphs G and G′ , respectively. Then G and G′ are isomorphic if and only if H and H ′ are isomorphic. Proposition 1 follows directly from Lemma 2. Lemma 2. Let H and H ′ be hexagon graphs of a cubic bridgeless graph. Let {h1 , . . . hk } and {h′1 , . . . h′k } be the set of hexagons of H and H ′ , respectively. Then {h1 , . . . hk } = {h′1 , . . . h′k }. Proof. For the sake of contradiction let us assume that {h1 , . . . hk } 6= {h′1 , . . . h′k }. Therefore, there exists a hexagon, say h ∈ {h1 , . . . hk } such that h is not in {h′1 , . . . h′k }. Let us suppose that h = {v0 , v1 , v2 , v3 , v4 , v5 } induces the local configuration depicted in Figure 1. By the assumption, we know that there exists an edge, say e, in the set of (red and blue) edges spanned by h (that is, in the hexagon spanned by h) that is a white edge with respect to the set of hexagons {h′1 , . . . h′k }. By the symmetry of the hexagon, we can assume that e is a blue edge, say e := v1 v0 . Let h′i and h′j , with i, j ∈ {1, . . . , k}, be the hexagons of {h′1 , . . . h′k } such that v0 ∈ h′i and v1 ∈ h′j ; thus, i 6= j. It implies that u3 , v5 , v3 ∈ h′i and w2 , v2 , v4 ∈ h′j (see Figure 1). Then, v3 has two neighbours in h′j , namely, v2 and v4 , a contradiction. 3 Rotation systems and embeddings without dual loops Recall that our goal in this section is to reformulate the following statement: every cubic bridgeless graph admits an embedding on a closed orientable surface without dual loops. For this purpose, we now introduce a combinatorial representation of embedding of graphs on closed orientable surfaces; namely rotation systems. Let G be a graph. For each v ∈ V (G), let πv be a cyclic permutation of the edges incident with v. A collection π = {πv : v ∈ V (G)} is called a rotation system of G. The proof of the following statement can be found in [9, §3.2]. Theorem 3. Let π be a rotation system of a graph G. Then π encodes an embedding of G on a closed orientable surfaces with set of face boundaries {e1 e2 · · · ek : ei = v i v i+1 ∈ E(G), πvi+1 (ei ) = ei+1 , ek+1 = e1 and k minimal}. (1) Moreover, the converse holds. That is, every embedding of G on a closed orientable surface defines a rotation system π of G where the set of face boundaries is given by the set described in (1). In Theorem 4, we state that blue perfect matchings of hexagon graphs of a cubic graph G define embeddings of G on closed orientable surfaces with distinguished set of face boundaries, and vice versa. The proof is based on a natural bijection between blue perfect matchings and rotation systems. We first need to make an observation. Observation 3. Let M be a blue perfect matching of H and let W be the set of white edges of H. Each cycle C in M ∪ W induces a subgraph in G defined by the set of edges {uv ∈ E(G) : euv ∈ C or e′uv ∈ C}. Theorem 4. Let G be a cubic graph, H be the hexagon graph of G and W be the set of white edges of H. Each blue perfect matching M of H encodes an embedding of G on a closed orientable surface with a set of face boundaries, the set of subgraphs of G induced by the cycles in M ∪ W . Moreover, the converse holds. That is, each embedding of G on a closed orientable surface defines a blue perfect matching M of H, where the set of subgraphs of G induced by all cycles in M ∪ W coincides with the set of face boundaries of the embedding. Proof. It suffices to prove that there is a bijective function f from the set of blue perfect matchings of H to the set of rotation systems of G such that for every blue perfect matching M of H, the set of subgraphs of G induced by the cycles in M ∪ W equals the set of subgraphs described in (1) defined by the rotation system f (M ) = π. Let v ∈ V (G), NG (v) = {u, w, z}, and without loss of generality (by Proposition 1) we assume that iv(u) = 0, iv(w) = 1 and iv(z) = 2. Let M be a blue perfect matching of H. The restriction of M to hv is either {v0 v1 , v2 v3 , v4 v5 } or {v1 v2 , v3 v4 , v5 v0 }. If the restriction is {v0 v1 , v2 v3 , v4 v5 }, then the cyclic permutation of the edges incident with v in the rotation system f (M ) = π of G is πv = (uv wv zv). Otherwise, the cyclic permutation is given by πv = (uv zv wv). It is a routine to check that f is the desired bijection. The following result is crucial for our approach. Proposition 5. Let G be a cubic graph, H be the hexagon graph of G, M be a blue perfect matching of H and W be the set of white edges of H. The embedding of G encoded by M has a dual loop if and only if there is a cycle in M ∪ W that contains the end vertices of a red edge. 4 Proof. An embedding of G has a dual loop if and only if there is an edge uv ∈ E(G) that belongs to exactly one face boundary, say C ′ . The face boundary C ′ is a subgraph of G induced by a cycle C of M ∪ W . We have C ′ is the only subgraph induced by a cycle of M ∪ W that contains uv if and only if euv and e′uv belong to C. The lemma follows. Motivated by Proposition 5, we shall say that a blue perfect matching M is safe if no cycle of M ∪ W contains the end vertices of a red edge. In Corollary 6 we establish the formulation of the DCDC Conjecture on hexagon graphs. Note that the result of Corollary 6 follows directly from Theorem 4 and Proposition 5. Corollary 6. A cubic graph G has a directed cycle double cover if and only if its hexagon graph H admits a safe perfect matching. 1.2 Braces A brace is a simple (that is, no loops and no multiple edges), connected, bipartite graph on at least six vertices, and with a perfect matching such that for every pair of nonadjacent edges, there is a perfect matching containing the pair of edges. In [7], McCuaig presented a method for generating braces. He showed that all braces can be constructed from a base set using four operations. In the following we describe McCuaig’s method for generating braces. Let H be a bipartite graph and x be a vertex of H of degree at least 4. Let N1 , N2 be a partition of NH (x) such that |N1 |, |N2 | ≥ 2. Let {x1 , v, x2 } be a set of vertices such that {x1 , v, x2 } ∩ V (H) = ∅. The expansion of x to x1 vx2 , or briefly an expansion of x is the operation composed of the following three steps: (i) delete x, (ii) add the new path x1 vx2 , and (3) connect every vertex of N1 (N2 , respectively) to the vertex x1 (x2 , respectively). For i ∈ {1, 2}, we say that Ni is the partition associated with xi . Note that if H ′ is a graph obtained from H by the expansion of a vertex, then H ′ is also bipartite. Augmentations. If H ′ is a bipartite graph obtained from H by adding a new edge, then we say that H ′ is obtained from H by a type-1 augmentation. Let x and w be two vertices in the same partition class of H such that x has degree at least 4. If H ′ is obtained from H expanding x to x1 vx2 and adding the new edge vw, then we say that H ′ is obtained from H by a type-2 augmentation. Let x and y be two vertices of H of distinct partition classes such that dH (x), dH (y) ≥ 4. Let H ′ be the bipartite graph obtained from H by expanding x and y to x1 vx2 and y 1 uy 2 respectively, and adding the new edge vu. If x and y are not connected in H, the operation for obtaining H ′ from H is called a type-3 augmentation, otherwise it is called a type-4 augmentation. x1 x w (a) type-1 augmentation x2 v w (b) type-2 augmentation Figure 2: Simple augmentations If H ′ is obtained from H by a type i augmentation for some i ∈ {1, 2, 3, 4}, then we say that H ′ is obtained from H by an augmentation. If i ∈ {1, 2}, then we say that H ′ is obtained from H by a simple augmentation (see Figure 2). Let B be the infinite set consisting of all bipartite Möbius ladders, ladders and biwheels (see Figure 3). 5 M6 M14 M10 L8 (a) Möbius ladders: M6 , M10 , M14 , M18 , . . . L12 L16 (b) Ladders: L8 , L12 , L16 , L20 , . . . B10 B12 B14 (c) Biwheels: B10 , B12 , B14 , B16 , . . . Figure 3: The base set B. Theorem 7 (McCuaig, 1998). Let H be a bipartite graph. Then H is a brace if and only if there exists a sequence H0 , H1 , . . . , Hk of bipartite graphs such that H0 ∈ B, Hi may be obtained from Hi−1 by an augmentation for each i ∈ {1, . . . , k} and Hk = H. 1.3 Main results The main results of this paper are the following. Theorem 8. Let G be a cubic graph. Then the hexagon graph H of G is a brace if and only if G is bridgeless. Proof. Let B, W , and R denote the set of blue, white, and red edges, respectively. Moreover, a blue edge is denoted by b, a white edge by w, and a red edge by r. Each pair of disjoint edges, {b, b′ }, {r, r′ }, or {b, r}, can be simply extended to a perfect matching of H. We note that each component of W ∪ R is a cycle on four vertices, a square. Let w, w′ be a pair of disjoint white edges. The edges w, w′ belong to the same square of W ∪ R, or to two different squares of W ∪ R. In either case w, w′ can be naturally extended to a perfect matching of H. Similarly, each edge of a pair w, r of disjoint white and red edges belongs to different squares of W ∪ R, and therefore it can be completed into a perfect matching of H. Finally we consider a pair b, w of disjoint white and blue edges. If the hexagon with b does not contain an end vertex of w, then it is not difficult to extend b, w to a perfect matching of H. Hence, let hu be the hexagon that contains b and an end vertex of w, and let hv be the hexagon that contains the other end vertex of w. Let b = ui ui+1 , w = uk vj , where i, j, k ∈ Z6 . If k ∈ / {i + 3, i + 4}, then b, w can be completed into a perfect matching of H that contains the edges b, w, and ui+3 ui+4 . Hence, without loss of generality we can assume that k = i + 3. Let euv = ui vj+3 and euz = ui+1 zl (notation as in (3) of Definition 1), where z is the neighbor of v in G such that the white edge with an end vertex ui+1 has an end vertex in hz , and l ∈ Z6 . Given that in G, edges uv, uz have a common end vertex u represented by hexagon hu , edge b = ui ui+1 can be seen as the transition between uv, uz, while uk uk+1 can be seen as this transition reversed. Now let G be bridgeless. We observe that two adjacent edges in a cubic bridgeless graph belong to a common cycle. Let C be such a cycle for uv, uz. The two possible orientations of C correspond to two disjoint cycles Cb , Cw in H, where b ∈ Cb and w ∈ Cw ; they contain the transition and transition reversed (between uv, uz), respectively. Let Mb be 6 the perfect matching of Cb consisting of all blue edges and Mw be the perfect matching of Cw consisting of all white edges. In particular, b ∈ Mb and w ∈ Mw . Since each hexagon of H is intersected by Cb ∪ Cw either in a pair of disjoint blue edges, or in the empty set, Mb ∪Mw can be extended to a perfect matching of H. On the other hand, if G has a bridge e = {u, v}, then let V1 be the component of G − e containing u. Any perfect matching of G extending b, w must induce a perfect matching of ∪x∈V1 hx \ {ui+3 }, but this set consists of an odd number of vertices and thus no perfect matching containing b, w can exist. Theorem 9. Let G be a cubic bridgeless graph and L8 denote the ladder on 8 vertices. There is a sequence H0 , H1 , . . . , Hk of bipartite graphs such that H0 = L8 , Hi can be obtained from Hi−1 by a simple augmentation for each i ∈ {1, . . . , k} and Hk is the hexagon graph of G. The crucial ingredients in the proof of Theorem 9 are odd ear decompositions of cubic bridgeless graphs. We now give a rough sketch of the proof. Let G be a cubic bridgeless graph, H be its hexagon graph, and (G0 , Gi , Pi )l be an odd ear decomposition of G (see Subsection 3.1). With each intermediate subgraph Gi of the odd ear decomposition of G we associate an auxiliary graph Hi′ . In particular, with (the cycle) G0 we associate the ladder L8 . For each i ∈ {1, . . . , l}, the auxiliary graph Hi′ contains the hexagons hv of H such that v has degree 3 in Gi . Hence, Hl′ contains all hexagons of H and indeed (by construction) it turns out to be isomorphic to H. The proof is based on the fact that for each ′ i ∈ {1, . . . , l}, it is possible to generate Hi′ from Hi−1 by a sequence of simple augmentations. The rest of the paper is devoted to prove Theorem 9. The proof of Theorem 9 is divided into two parts. The first part is the generation of hexagon graphs from square graphs and the second is the construction of square graphs from the ladder on 8 vertices. In Section 2, we introduce the concept of square graphs and prove that hexagon graphs can be obtained from square graphs by a short sequence of simple augmentations. Section 3 and Section 4 focus on the construction of square graphs. 2 Square graphs A square is a complete bipartite graph on 4 vertices, namely K2,2 . We say that a bipartite graph has a square s if it contains s as a subgraph. Next we define square graphs. Definition 2 (Square graphs). Let G be a cubic bridgeless graph with vertex set V and edge set E. Let M be a perfect matching of G. An M -square graph of G is a bipartite graph Q with neither loops nor multiple edges satisfying the following properties: 1. For each vertex v in V, the graph Q has a square sv . If u, v ∈ V are such that u 6= v, then sv and su are vertex disjoint subgraphs of Q. Moreover, V (Q) = {V (v) : v ∈ V }. 2. The set of edges of Q is given by E(Q) = {E(sv ) : v ∈ V } ∪ {uv : uv ∈ E}, where {uv : uv ∈ E} is defined such that the following conditions hold: (a) For each edge uv ∈ E, there are edges eu in E(su ) and ev in E(sv ) such that the subgraph of Q induced by the set of edges {eu , ev } ∪ uv is isomorphic to K2,2 . In particular, |uv| = 2. The edges eu and ev are called the supporting edges of uv in su and sv , respectively. (b) Let v ∈ V and NG (v) = {u, w, z}. If uv ∈ M , then the supporting edges of wv and zv in sv are vertex disjoint. 7 We say that sv is the square associated with vertex v and that {sv : v ∈ V } is the set of squares of Q. For each uv ∈ E, if uv ∈ M , then we say that (su , sv ) is a pair of matched squares of Q. Moreover, the subset of edges uv is called the projection of uv in Q. We usually denote by {vi : i ∈ Z4 } the vertex set of the square sv and by {vi vi+1 : i ∈ Z4 } its edge set. Note that the graph obtained by contracting each square of Q to a single point and then by deleting multiple edges is precisely G. The following is a natural observation about square graphs. Observation 4. For every connected component C of G − M (C is a cycle since G is cubic), there exists a ladder L on 4 · |C| vertices in the set of connected components of Q − {e : e ∈ M } such that v is a vertex of C if and only if sv is a square of L. In Lemma 10, we state that hexagon graphs can be generated from square graphs using simple augmentations. Lemma 10. Let G be a cubic bridgeless graph, M be a perfect matching of G and Q be an M -square graph of G. Then there is a sequence of bipartite graphs H0 , H1 , . . . , Hl such that H0 = Q, Hi may be obtained from Hi−1 by a simple augmentation for each i ∈ {1, . . . , l} and Hl is the hexagon graph of G. Proof. We first describe an operation composed of a sequence of simple augmentations which we apply to each pair of matched squares in order to generate a pair of hexagon-neighbors; we shall call this operation a double augmentation. Let (su , sv ) be a pair of matched squares of Q. By definition, all distinct configurations of the supporting edges of uv in su and sv , respectively, are the ones depicted in Figure 4. sv su (a) su sv (b) sv su (c) Figure 4: Possible locations of the supporting edges of uv in su and sv for a pair (su , sv ) of matched squares of Q. Supporting edges are depicted by thick lines. We assume that the supporting edges of uv for the pair (su , sv ) are configured as in Figure 4(a). Consider the vertex labeling depicted in Figure 5(a). Next, we describe the aforementioned operation with input the pair (su , sv ). Double augmentation on (su , sv ): (see Figure 5) [step 0:] addition of the two new edges u1 v0 and u2 v3 . [step 1:] expansion of v0 to v01 vv02 in such a way that the partition associated with v02 is {u1 , u3 } and with v01 is {v1 , v3 , z1 } and addition of the new edge vv2 . [step 2:] addition of the new edge vu2 . [step 3:] expansion of u2 to u12 uu22 in such a way that the partition associated with u12 is {u1 , x2 , u3 } and with u22 is {v1 , v, v3 } and addition of the new edge uv02 . [step 4:] addition of the new edge uu0 . We observe that in steps 1 and 3 respectively, expansion of v0 and expansion of u2 respectively are allowed given that the degrees are 5 and 6 respectively; recall that degree at least 4 is required for expansion; see Subsection 1.2. In case that the supporting edges of uv for the pair (su , sv ) are configured as in Figure 4(b) or as in Figure 4(c) respectively (set the same vertex labeling), if we replace the edges added at the step 0 of the double augmentation described above by u2 v3 , u3 v0 and u2 v1 , u3 v0 respectively, then the local 8 configuration obtained is the one depicted in Figure 5(b). Therefore, if we continue applying steps 1, 2, 3 and 4 as before we obtain the local configuration depicted in Figure 5(f). x1 w1 x2 u2 u1 w2 v1 v2 x1 u2 u1 sv su v3 y1 z2 v3 y1 (a) Initial configuration x1 w1 x2 u2 u1 u3 z1 x1 z2 v2 u1 v3 u0 u1 2 u u2 2 v2 y1 v3 1 v0 v z1 y2 z2 (c) step 1 w1 x2 w2 v1 u0 (b) step 0 w2 v1 u2 u1 v0 y2 w1 x2 2 v0 u0 u3 z1 v2 x1 sv v0 y2 w2 v1 su u0 u3 w1 x2 w2 v1 x1 v2 u1 v3 u0 w1 x2 u1 2 u u2 2 w2 v1 v2 2 v0 u0 u3 y1 y2 1 v0 v z1 (d) step 2 u3 z2 y1 1 v0 2 v v0 z1 y2 u3 z2 y1 (e) step 3 y2 v3 1 v0 2 v v0 z1 z2 (f) step 4 Figure 5: Double Augmentation on (su , sv ). In subfigure (f), red edges are depicted by red lines. We claim that the graph obtained from Q by performing a double augmentation on every pair of matched squares is a hexagon graph of G. The disjoint subsets of vertices {u0 , u3 , u1 , u12 , u, v02 } and {v1 , v2 , v3 , v01 , v, u22 } induce hexagons. Let hu and hv denote them respectively. The claim follows by setting {u0 u3 , u1 u12 , uv02 } and {v1 v2 , v3 v01 , vu22 } to be the subsets of red edges in hu and hv , respectively (see Figure 5(f)). To conclude, since steps 0, 2 and 4 correspond to type-1 augmentations, and steps 1 and 3 correspond to type-2 augmentations, we have that a double augmentation on a pair of matching related squares is composed of a sequence of simple augmentations. 3 Construction of square graphs In order to prove Theorem 9, by Lemma 10 it suffices to show that we can construct an M -square graph of G, for some perfect matching M of G, from the ladder on 8 vertices using simple augmentations. In this section we develop a method to construct square graphs following an ear decomposition of the underlying cubic bridgeless graph G and using simple augmentations. 3.1 Odd ear decomposition of a cubic bridgeless graph Let G be a graph. We say that a path, or a cycle of G, is even (odd respectively) if it has an even (odd respectively) number of edges. An odd ear decomposition of G, denoted by (G0 , Gi , Pi )l , consists of a sequence of subgraphs G0 , G1 , . . . , Gl and a sequence of odd paths P1 , . . . , Pl of G such that G0 is an even cycle of G, Gl = G and for each i ∈ {1, . . . , l} the subgraph Gi is obtained from Gi−1 joining two vertices αi and βi in V (Gi−1 ) by a path Pi , where Pi is such that V (Pi ) ∩ V (Gi−1 ) = {αi , βi } and 9 E(Pi ) ∩ E(Gi−1 ) = ∅. It is folklore that every edge of a cubic bridgeless graph is contained in a perfect matching and hence, the class of cubic bridgeless graph is a subclass of the class of 1-extendable graphs. In addition, every 1-extendable graph admits an odd ear decomposition [6, §5.4]. Let G be a cubic bridgeless graph and (G0 , Gi , Pi )l be an odd ear decomposition of G. We say that a perfect matching M of G is absolute in (G0 , Gi , Pi )l if the restriction of M to E(Gi ) is a perfect matching of Gi for every i ∈ {0, 1, . . . , l}. The next observation is straightforward. Observation 5. For every odd ear decomposition (G0 , Gi , Pi )l of a cubic bridgeless graph G, there exists a perfect matching M of G that is absolute in (G0 , Gi , Pi )l . In the rest of the paper, we deal only with perfect matchings that are absolute in a given odd ear decomposition (G0 , Gi , Pi )l . Let i ∈ {1, . . . , l} and let Vj (Gi ) denote the subset of vertices of V (Gi ) that have degree j in Gi for each j ∈ {2, 3}. Let u, v ∈ V3 (Gi ) and P be a path of Gi with end vertices u, v such that V (P ) ∩V3 (Gi ) = {u, v}. In other words, every inner vertex of P belongs to V2 (Gi ). We say that P is a (u, v)-path of Gi and usually denote P by p(u, v). Note that there may exist multiple (u, v)-paths. We shall denote by P(Gi ) the set of all (u, v)-paths for all u, v in V3 (Gi ). We note that if v is a vertex in V3 (Gi ), then there are three (not necessarily distinct) vertices x, y, z in V(Gi ), such that p(x, v), p(y, v), p(z, v) ∈ P(Gi ). We say that the set {x, y, z} is the set of pseudoneighbors of v in Gi . Observe that if M is a perfect matching of G and vw ∈ M , then there is a unique path P ∈ {p(x, v), p(y, v), p(z, v)} such that vw ∈ E(P ). We refer to P as the matching-path of v in Gi (with respect to M ). If vw is not in E(P ), then P is called a cycle-path of v in Gi . Note that a path p(u, v) in P(Gi ) could be both, a matching-path of v and a cycle-path of u. However, since M is a perfect matching that is absolute in (G0 , Gi , Pi )l , the path Pi = p(αi , βi ) ∈ P(Gi ) is always a cycle-path of both αi and βi in Gi (see Figure 9(a)). In Subsection 3.2, we generalize the definition of square graphs of a cubic graph G to the intermediate graphs G0 , G1 , . . . , Gl associated with an odd ear decomposition of G. 3.2 Ear square graphs In this section and in the rest of the paper, G is a cubic bridgeless graph, (G0 , Gi , Pi )l is an odd ear decomposition of G and M is a perfect matching of G that is absolute in (G0 , Gi , Pi )l . Definition 3 (Ear square graphs). For each i ∈ {1, . . . , l}, a (Gi , M )-ear square graph is a bipartite graph Qi with neither loops nor multiple edges that satisfies the following properties: 1. For each vertex v in V3 (Gi ), the graph Qi has a square sv . For every u,v in V3 (Gi ) with u 6= v, the squares sv and su are vertex disjoint subgraphs of Qi . Moreover, V (Qi ) = {V (sv ) : v ∈ V3 (Gi )}. 2. The set of edges of Qi is given by {E(sv ) : v ∈ V3 (Gi )} [ ˙ p(u, v) p(u,v)∈P(Gi ) where {p(u, v) : p(u, v) ∈ P(Gi )} is defined such that the following conditions hold: (a) For each p(u, v) ∈ P(Gi ), we have |p(u, v)| = 2, and there are edges eu in E(su ), ev in E(sv ) such that the subgraph of Qi induced by the set of edges {eu , ev }∪p(u, v) is isomorphic to K2,2 . The edges eu and ev are called the supporting edges of p(u, v) in su and sv , respectively. 10 (b) Let v be a vertex in V3 (Gi ) and {x, y, z} be its set of pseudo-neighbors. If p(x, v) is the matching-path of v in Gi , then the supporting edges of p(v, y) and p(v, z) in sv are vertex disjoint (see Figure 6). (c) Elements in {p(u, v) : p(u, v) ∈ P(Gi )} are pairwise disjoint. x sx sx p(v, x) sx sx p(v, x) p(v, x) p(v, x) sv v p(v, z) p(v, y) sv y p(v, y) p(v, z) (a) (b) sv sy sz sz p(v, y) sv sy z p(v, z) p(v, y) p(v, z) sy sz (c) sy sz (d) (e) Figure 6: Local representation of Gi and a (Gi , M )-ear square graph of Gi . In subfigure (a), we depict a vertex v ∈ V3 (Gi ) with x, y, z ∈ V3 (Gi ) its pseudo-neighbors and p(v, x) the matching-path of v in Gi . Dashed edges represent edges from M . In subfigures (b)-(e), we depict all the allowed locations of the supporting edge of p(v, x) in sv in a (Gi , M )-ear square graph of Gi . In each subfigure the supporting edge is depicted by a thicker line. For every p(u, v) ∈ P(Gi ), the set p(u, v) is said to be its projected (u, v)-path in Qi . If p(u, v) is the matching-path of v in Gi , we say that p(u, v) is the projected matching-path of sv in Qi . Since V3 (Gl ) = V (G), the following proposition follows from Definition 2 and Definition 3. Observation 6. A graph H is a (Gl , M )-ear square graph if and only if H is an M -square graph. In Lemma 11 we formalize the construction of square graphs using ear square graphs and simple augmentations. This lemma is proved in Section 4. Lemma 11 (Construction of square graphs). Let G be a cubic bridgeless graph, (G0 , Gi , Pi )l be an odd ear decomposition of G and M be a perfect matching of G that is absolute in (G0 , Gi , Pi )l . Let L8 denote the ladder on 8 vertices (see Figure 3(b)). The following two properties hold. 1. A (G1 , M )-ear square graph Q1 can be generated from L8 using type-1 augmentations. 2. Let i ∈ {2, . . . , l} and Qi−1 be a (Gi−1 , M )-ear square graph. Then a (Gi , M )-ear square graph Qi can be generated from Qi−1 using a sequence of simple augmentations. Note that Lemma 11 along with Observation 6 and Lemma 10 imply Theorem 9. 4 Proof of Lemma 11 In this section, G is a cubic bridgeless graph, (G0 , Gi , Pi )l is an odd ear decomposition of G and M is a perfect matching of G that is absolute in (G0 , Gi , Pi )l . Let L8 denote the ladder on 8 vertices. Moreover, for each i ∈ {1, . . . , l}, let Qi denote a (Gi , M )-ear square graph. In what follows we enunciate two natural properties about ear square graphs. The result of Proposition 12 follows directly from Definition 3. Proposition 12. For every i ∈ {1, . . . , l}, each square sv in Qi with V (sv ) = {vj : j ∈ Z4 } is such that there exists a unique j ∈ Z4 such that dvj = dvj+1 = 4 and dvj+2 = dvj+3 = 3. 11 Proposition 13. Let p(x, y) and p(w, z) be paths in P(Gi ). Let p(x, y) be the projected path of p(x, y) and p(w, z) be the projected path of p(w, z) in Qi . Then the subgraph S of Qi with a set of edges p(x, y) ∪ p(w, z) ∪ E(sx ∪ sy ∪ sw ∪ sz ) is isomorphic to one of the 9 graphs (configurations) depicted in Figure 7. w2 x2 x1 y2 y1 w1 z1 w2 z2 x2 y2 (a) Configuration 1 z2 =y2 x2 x1 =w1 y1 z1 (e) Configuration 5 z2 x1 =w1 z1 w1 x1 w2 y1 z2 y2 z1 (f) Configuration 6 y2 y1 (b) Configuration 2 w2 x2 y1 x2 x1 =w1 w1 x2 =w2 z1 z2 y2 (c) Configuration 3 x1 w2 y1 z2 z1 (d) Configuration 4 x2 w1 =x1 w2 y1 y2 z2 z1 (g) Configuration 7 x1 =w1 x2 =w2 y1 y2 x1 =w1 x2 =w2 y1 =z1 y2 =z2 z2 z1 (h) Configuration 8 (i) Configuration 9 Figure 7: In (a) is depicted the unique subgraph that arises when vertices x, y, z, w ∈ V3 (Gi ) are all distinct. From (b) to (d) the three possible subgraphs that arise when |{sx , sy , sw , sz }| = 3. Figures from (e) to (i) depict all the possible situations when |{sx , sy , sw , sz }| = 2. Proof. We first suppose that |{x, y, w, z}| = 4. Then x, y, z, w ∈ V3 (Gi ) are all distinct and the squares sx , sy , sw , sz in Qi are vertex disjoint. Therefore, S is isomorphic to the graph depicted in Figure 7(a). We now suppose that |{x, y, w, z}| = 3. It means that the paths p(x, y) and p(w, y) have one common end vertex. Without loss of generality we suppose that x = w, and then, sx , sy and sz are vertex disjoint. In the subgraph S three distinct situations depending on the location of the supporting edges ex and ew of p(x, y) and p(w, z) in sx can arise: a.1) either |ex ∩ ew | = 1, or a.2) ex = ew , or a.3) ex ∩ ew = ∅. If situation a.1) holds, then S is isomorphic to configuration 2, see Figure 7(b). If situation a.2) holds, then S is isomorphic to configuration 3, see Figure 7(c), and if situation a.3) holds, then S is isomorphic to configuration 4, see Figure 7(d). 12 We finally suppose that |{x, y, w, z}| = 2. Without loss of generality we assume that x = w and y = z. If p(x, y) = p(w, z), then S is isomorphic to the graph depicted in Figure 7(i), this graph is called configuration 9. Otherwise, in the graph S several distinct situations depending on the location of the supporting edges ex and ew of p(x, y) and p(w, z) in sx and of the supporting edges ey and ez of p(x, y) and p(w, z) in sy may arise: b.1) either |ex ∩ ew | = 1 and |ey ∩ ez | = 1, or b.2) ex ∩ ew = ∅ and ey ∩ ez = ∅, or b.3) |ex ∩ ew | = 1 and ey ∩ ez = ∅, or b.4) ex = ew and ey ∩ ez = ∅, or b.5) |ex ∩ ew | = 1 and ey = ez , or ex = ew and ey = ez . If situation b.1), b.2), b.3) or b.4) holds, then S is isomorphic to configuration 5, 6, 7, or 8, respectively. Those configurations are depicted in Figure 7). Situations described in b.5) do not occur in Qi given that Qi does not have multiple edges. We clarify the last statement in the following paragraph. If we suppose that |ex ∩ ew | = 1 or ex = ew then, there exists a vertex, without loss of generality we assume that such a vertex is x1 ∈ ex ∩ ey such that x1 y1 and x1 z1 are edges of S. We recall that Qi does not have multiple edges. Since ey = ez , we have y1 = z1 and S has a double edge, a contradiction. 4.1 Generating ear square graphs This section is devoted to prove Lemma 11. Proof of Lemma 11, part 1 We need to prove that we can generate a (G1 , M )-ear square graph from L8 using type-1 augmentations (addition of new edges). We consider the vertex-labeling of L8 depicted in Figure 8(a). Let p(u, v), p(x, y) and p(w, z) be the only three paths in P(G1 ), where u = x = w, v = y = z and P1 = p(u, v). We have that G1 satisfies one of the following properties: 1) either p(x, y) is the matching-path of v and of u in G1 , or 1′ ) p(w, z) is the matching-path of v and of u in G1 , or 2) p(x, y) is the matching-path of v in G1 and p(w, z) is the matching-path of u in G1 , or 2′ ) p(w, z) is the matching-path of v in G1 and p(x, y) is the matching-path of u in G1 . By symmetry, it suffices to prove that for each i ∈ {1, 2}, we can generate from L8 a (G1 , M )-ear square graph, where G1 and M satisfies i). We first claim that if 1) holds, then the bipartite graph obtained from L8 by adding the new edges v0 u2 and v3 u1 is a (G1 , M )-ear square graph (see Figure 8(b)). The validity of this claims follows from considering {v0 u2 , v3 u1 } to be the projected path of p(x, y), {v2 u2 , v3 u3 } to be the projected path of p(w, z) and {v0 u0 , v1 u1 } to be the projected path of p(u, v). Secondly, we claim that if 2) holds, then the bipartite graph obtained from L8 by adding the new edges v1 u3 and v0 u2 is a (G1 , M )-ear square graph (see Figure 8(c)). In this case, if we let {v0 u2 , v3 u1 } be the projected path of p(u, v), {v2 u2 , v3 u3 } be the projected path of p(w, z) and {v0 u0 , v1 u1 } be the projected path of p(x, y), then the claim follows. 13 v3 v0 v3 v0 L8 p(x, y) sv p(u, v) v2 u1 u2 p(u, v) su u0 Q1 sv v1 v2 v1 v2 u1 u2 u1 u2 su u0 (a) A ladder L8 on 8 vertices p(w, z) sv v1 p(u, v) u3 v3 v0 Q1 p(w, z) su p(x, y) u3 u0 (b) G1 -ear square graph p(u, v) u3 (c) G1 -ear square graph Figure 8: Generation of Q1 from L8 . Proof of Lemma 11, part 2 For each i ∈ {2, . . . , l}, we need to show that from a (Gi−1 , M )-ear square graph we can generate a (Gi , M )-ear square graph using simple augmentations. For this purpose, the idea is to make local changes; we basically replace the projected paths in Qi−1 of the paths that contain αi and βi by two new squares sαi , sβi , and by the new projected paths incident with them. Moreover, we modify neither any square in Qi−1 , nor the position of the supporting edges of the projected paths incident with them (see Figure 9). Here, αi , βi denote the end vertices of the path Pi from (G0 , Gi , Pi )l . Let p(x, y) and p(w, z) be the projected paths in Qi−1 such that αi belongs to V (p(x, y)) in Gi−1 and βi belongs to V (p(w, z)) in Gi−1 . Qi Gi Qi−1 x Gi−1 y Qi−1 sx sαi αi p(x, y) sy sỹ y ỹ sx ′ sy sỹ sy ′ sy ′ sw sw Pi w βi z p(w, z) sz sz sβ (a) Pi =: αi · · · βi is a cycle-path of αi and βi in Gi . Paths p(x, y) and p(w, z) contain αi and βi in Gi−1 . (b) p(x, y) and p(w, z) are the projected paths of p(x, y) and p(w, z) in Qi−1 . i (c) (Gi , M )-ear square graph. Squares sαi and sβi are constructed and also the projected paths incident with them. Figure 9: p(x, y) and p(w, z) are the projected paths in Qi−1 such that αi ∈ V (p(x, y)) and βi ∈ V (p(w, z)) in Gi−1 . In (a), dashed edges represent the perfect matching M that is absolute in (G0 , Gi , Pi )l . In (b)-(c), dashed lines represent projected matching-paths. In what follows, for the sake of simplicity we set u = αi and v = βi . We attempt to generate the two new squares su and sv and the projected paths p(u, x), p(u, y), p(u, v), p(v, z) and p(v, w). In order to cover all cases we need to take care of two issues, first the interaction between the projected paths p(x, y) and p(w, z) in Qi−1 , which is described by Proposition 13 and depicted in Figure 7, and the second issue is the location of the perfect matching M with respect to the edges and paths incident with u and v. 14 For the second issue, we know that Pi is a cycle-path ( recall that since M is a perfect matching that is absolute in (G0 , Gi , Pi )l , the path Pi = p(αi , βi ) ∈ P(Gi ) is always a cycle-path of both u and v in Gi — see Figure 9(a)), and therefore if p(x, y) 6= p(w, z), then the matching-path of u is either p(u, y) or p(u, x) and the matching-path of v is either p(v, w) or p(v, z). In Figure 10, we describe the cases and depict examples for the situation that x = w and y 6= z. The remaining cases, for example when all x, w, y, z are different, are analogous. x x x x u v u v u v u v y z y z y z y z (a) instance i): p(u, y) matching-path of u and p(v, z) matching-path of v. (b) instance ii): p(u, x) matching-path of u and p(v, z) matching-path of v. (c) instance iii): p(u, y) matching-path of u and p(v, w) matching-path of v. (d) instance iv): p(u, x) matching-path of u and p(v, w) matching-path of v. Figure 10: In the subfigures we depicted an example of each instance when the paths p(x, y) and p(w, z) in Gi−1 intersect in one vertex (see Configurations 2,3 and 4 in Figure 7). Bold edges represent the perfect matching. Moreover, p(u, v) = Pi . In the case that p(x, y) = p(w, z), without loss of generality we can assume that u and v are placed (with respect to x and y) as depicted in Figure 11. Then, we have that the matching-path of u is either p(u, x) or p(u, v) (with p(u, v) 6= Pi ) and the matching-path of v is either p(v, y) or p(u, v) (with p(u, v) 6= Pi ). In Figure 11, we describe these situations with a corresponding example. x x x x u u u u Pi Pi Pi Pi v v v v y y y y (a) instance i′ ): p(u, x) matching-path of u and p(v, y) matching-path of v. (b) instance ii′ ): p(u, x) matching-path of u and p(u, v) matching-path of v. (c) instance iii′ ): p(u, y) matching-path of u and p(v, w) matching-path of v. (d) instance iv′ ): p(u, v) matching-path of u and v. Figure 11: In each figure is depicted an example of the distinct instances in the case that p(x, y) = p(w, z) (see configuration 9 in Figure 7). Bold edges represent the perfect matching. Summarizing, to prove Lemma 11.2, it suffices to prove that from each configuration of the projected paths p(x, y) and p(w, z) it is possible to generate all instances i), ii), iii) and iv) in case that p(x, y) 6= p(w, z) and that it is possible to generate all instances i′ ), ii′ ), iii′ ) and iv′ ) in case that p(x, y) = p(w, z). Before we go into the analysis of the configurations we shall present an operation consisting of a sequence of simple-augmentations that we constantly use in order to construct two new squares; we shall call this operation a basic square construction. This operation is very useful and crucial to reduce the number of cases. 15 The input of the basic square construction is the bipartite subgraph graph depicted in Figure 12(a) with distinguished edges ea , eb , ec and ed . The output is the bipartite subgraph depicted in Figure 12(c) with distinguished edges ea , e′b , e′c and e′d . In Figure 12, we depict the sequence of simple-augmentations that compose the basic square construction. It is clear that if H is the graph obtained by applying the basic square construction in a subgraph of a brace, then H is a brace. In what follows, we constantly use this operation and the latter remark. ea ec eb ed ea e′b type-2 type-2 augmentation augmentation (a) ec ea e′c ed e′b ed type-2 augmentation ea e′b type-2 augmentation (b) e′c e′d ea e′c e′b e′d type-1 augmentation 1 (c) Figure 12: Basic square construction. In subfigure (a) the input subgraph with distinguished edges ea , eb , ec , ed , in subfigure (b) the steps of the basic square construction and in subfigure (c) the output subgraph with distinguished edges ea , e′b , e′c , e′d , Let V (sx ) = {xj : j ∈ Z4 }, V (sy ) = {yj : j ∈ Z4 }, V (sw ) = {wj : j ∈ Z4 }, V (sz ) = {zj : j ∈ Z4 }. Without loss of generality we assume that p(x, y) = {x1 y1 , x2 y2 }, p(w, z) = {w1 z1 , w2 z2 }, and that x1 , w1 , y2 , z2 are in the same partition class, depicted in black in Figure 7. We first study configurations 2, 5, and 7, then 3 and 8, afterwards configurations 4 and 6, and finally configurations 1 and 9. Configurations 2, 5 and 7 are respectively depicted in Figures 7(b), 7(e) and 7(g). These configurations have a common property, namely: with the notation of Figure 7 each of these configurations may be obtained from Figure 13(a) by possible identifying y2 , z2 (case of Configuration 5). Next we show how to generate instances i), ii), iii) and iv) of Figure 10. Generation of instances i) and iv): Let Q2i−1 be the graph obtained from Qi−1 by expanding the vertex x1 to x11 v1 x21 in such a way that the partition associated with the vertex x21 is either {x2 , z1 } if we are generating instance i), or {w2 , z1 } if we are generating instance iv). Then, we add the new edge v1 z2 if we are generating instance i), or v1 y2 if we are generating instance iv) —see Figures 13(b) and 13(e) without the bold edges for an ilustration of Q2i−1 in each case. Then, we 2 consider the graph Q2,1 i−1 obtained from Qi−1 by adding the bold edge. In Figures 13(b) and 13(e), 2,1 the graph Qi−1 is locally depicted for each case. We get the desired instances by applying the basic square construction. We describe this in more details. For instance i): with the notation of Figure 12(a) and 13(b), it is enough to consider ea = x11 w2 , eb = z1 z2 , ec = x11 x2 and ed = y1 y2 . For instance iv): with the notation of Figure 12(a) and 13(b), it is enough to consider ea = z1 z2 , eb = x11 w2 , ec = y1 y2 and ed = x11 x2 . For generating instances ii) and iii): Let Q2i−1 be the graph obtained from Qi−1 by expanding the vertex x1 to x11 v1 x21 in such a way that the partition associated with the vertex x21 is {y1 , z1 }. Then, we add the new edge v1 z2 if we are generating instance ii), or v1 y2 if we are generating instance iii). 2 1 Then, we consider the graph Q2,1 i−1 obtained from Qi−1 by adding the new edge x1 y1 if we are generating instance ii) or x11 z1 if we are generating instance iii). In Figures 13(c) and 13(d), the 16 graph Q2,1 i−1 is locally depicted for each case. For instance ii): with the notation of Figure 12(a) and 13(b), it is enough to consider ea = x11 w2 , eb = z1 z2 , ec = y1 y2 and ed = x11 x2 . For instance iii): with the notation of Figure 12(a) and 13(b), it is enough to consider ea = x11 x2 , eb = y1 y2 , ec = x11 w2 and ed = z1 z2 . x2 w2 x2 x1 y2 y1 (a) z1 z2 y2 (b) w2 x1 1 y1 w2 x1 1 x2 w2 x1 1 x2 w2 x1 1 v1 v1 v1 v1 x2 1 x2 1 x2 1 x2 1 z1 2,1 Qi−1 x2 z2 for i) y2 (c) y1 Q2,1 i−1 z1 z2 for ii) y2 y1 (d) 2,1 Qi−1 z1 z2 for iii) y2 y1 (e) Q2,1 i−1 z1 z2 for iv) Figure 13: Local view of Q2,1 i−1 in the generation of instances i), ii), iii) and iv) for configurations 2, 5 and 7. In each configuration, we may possible have y2 = z2 . In (a), for obtaining Configurations 3 and 8 it is enough to identify x2 and w2 , and delete multiple edges. Configurations 3 and 8 are respectively depicted in Figures 7(c) and 7(h). Note that with the notation of Figure 7, both configurations may be obtained from Figure 13(a) by identifying x2 , w2 and by removing the double edge. Therefore, the reasoning for configurations 2, 5 and 7 applies also for configurations 3 and 8. Configurations 4 and 6. These configurations are respectively depicted in Figures 7(d) and 7(f). With the notation of Figure 7, both configurations can be locally depicted as in Figure 14(a). Moreover, using the symmetry of both configurations 4 and 6, without loss of generality we can assume that either the degree of x1 and w2 in Qi−1 is 4 or the degree of x1 and x2 in Qi−1 is 4. Therefore, in either case we are allowed to expand x1 . Next we show how to generate each instances i), ii), iii) and iv) of Figure 10. Generation of instance i): Let Q2i−1 be the graph obtained from Qi−1 by expanding the vertex x1 to x11 u1 x21 in such a way that the partition associated with the vertex x21 is {w2 , y1 }. Then, we add the new edge u1 y2 . Consider the Figure 14(b) without the bold edge for a local ilustration of Q2i−1 . 1 2 Then, we consider the graph Q2,1 i−1 obtained from Qi−1 by adding the new edge x1 w2 , namely, the bold edge of Figure 14(b). We finally obtain the desired instance i) by applying the basic square construction in the same fashion as for the case of Configurations 2, 5, and 7. Generation of instances ii) and iv): Let Q2i−1 be the graph obtained from Qi−1 by expanding the vertex x1 to x11 u1 x21 in such a way that the partition associated with the vertex x21 is {x2 , y1 }. Then we add the new edge u1 w1 if we are generating instance ii), or u1 z2 if we are generating instance 2,1 iv). Then, we consider the graph Qi−1 obtained from Q2i−1 adding the new edge x11 x2 (bold edge in Figures 14(c) and 14(e)). In Figures 14(c) and 14(e), the graph Q2,1 i−1 for the generation of both instances is locally depicted. Again, we get the desired instances ii) and iv) by applying the basic square construction in the same fashion as for the case of Configurations 2, 5, and 7. Generation of instance iii): If the edge x2 z2 ∈ / E(Qi−1 ), then add x2 z2 . We denote by Q1i−1 either the graph obtained from Qi−1 by adding x2 z2 or, the graph Qi−1 such that x2 z2 ∈ E(Qi−1 ). Hence, z2 , y2 are neighbors of x2 in Q1i−1 and clearly y2 6= z2 (see Figures 7(d) and 7(f)). Let Q2,1 i−1 be the 1 2 1 graph obtained from Qi−1 by expanding the vertex x2 to x2 u1 x2 in such a way that the partition ∗ associated with the vertex x22 is {z2 , y2 }. Then, we add the new edge u1 y1 . Then we obtain Q2,1 i−1 in 17 x2 x1 w1 x2 w2 x1 1 w1 x2 w2 x1 1 u1 y1 (a) z2 z1 y2 (b) y1 x1 w2 z2 z1 y2 for i) (c) x2 w2 x1 1 y1 z2 2,1 Qi−1 z1 for ii) w1 w2 u1 x2 2 x2 1 Q2,1 i−1 w1 u1 u1 x2 1 y2 x1 2 w1 x2 1 y2 (d) y1 ∗ Q2,1 i−1 z2 z1 for iii) y2 (e) y1 2,1 Qi−1 z2 z1 for iv) ∗ 2,1 Figure 14: Local view of Q2,1 i−1 or Qi−1 in the generation of instances i), ii), iii) and iv) for configurations 4 and 6. In each configuration, we have that y2 6= z2 and y1 6= z1 . In case (d), the edge x12 z2 may exist. ∗ 2,1 the following way: if the edge x2 z2 ∈ E(Qi−1 ), then we obtain Qi−1 from Q2,1 i−1 by adding the new ∗ 2,1 2,1∗ edge x12 z2 . Otherwise, Q2,1 = Q . In Figure 14(d) the graph Q is locally depicted. Again, i−1 i−1 i−1 we use the basic square construction to complete the generation. Configuration 1. This configuration is depicted in Figure 7(a). By the symmetry of configuration 1 it suffices to show that we can generate instance iii); this can be generated in the same fashion as the previous instance iii) for configurations 4 and 6. Configuration 9 is depicted in Figure 7(i). To make things easier, we depict in Figure 15 the subgraphs that we want to generate from Configuration 9; they correspond to the instances i′ ), ii′ ), iii′ ) and iv′ ) of Figure 11. x1 y1 x2 x1 x2 x1 x2 su su su sv sv sv y2 (a) Instance i’) y1 y2 (b) Instance ii’) and iv’) y1 y2 (c) Instance iii’) and iv’) Figure 15: Instances i’), ii’), iii’) and iv’) of Figure 11 for configuration 9. We first focus on the generation of the configurations depicted in Figure 15(b) and Figure 15(c). By symmetry, it suffices to generate only one of them, say we generate the configuration depicted in Figure 15(c). We split this case into two subcases: (*) at least one vertex of {x1 , x2 } has degree 4 in Qi−1 and (**) x1 and x2 have degree 3 in Qi−1 . subcase (*): without loss of generality we assume that x1 has degree 4 in Qi−1 . Consider the 1 2 graph Q2,1 i−1 obtained from Qi−1 by expanding x1 to x1 ux1 in such a way that the partition associated with the vertex x21 is {x2 , y1 }. Then we add the new edges uy2 and x2 x11 . Next, 2,1 2 1 we consider the graph Q2,1,2 i−1 obtained from Qi−1 by expanding y2 to y2 vy2 in such a way that the partition associated with the vertex y22 is {u, x2 }. Then we add the new edge vx21 . 18 1 ′ 2 Furthermore, let Q2,1,2,2 be the graph obtained from Q2,1,2 i−1 i−1 by expanding x2 to x2 u x2 in such a way that the partition associated with the vertex x22 is {y22 , x21 }. Then we add the new edge uu′ . We finally consider the graph Q2,1,2,2,2,1 obtained from Q2,1,2,2 by expanding u to u1 wu2 i−1 i−1 in such a way that the partition associated with the vertex u2 is {u′ , x11 }. Then we add the new edges wx22 and wv. The graph Q2,1,2,2,2,1 is locally equal to the subgraph depicted in i−1 Figure 15(c). subcase (**): we recall that the set of vertices of the square sx is given by {xi : i ∈ Z4 }. Then, by Proposition 12 the vertices x0 and x3 have degree 4. Let Q2,1 i−1 be the graph obtained from 1 2 Qi−1 by expanding x3 to x3 ux3 in such a way that the partition associated with the vertex x23 is {x0 , x2 }. Then we add the new edges ux1 and x13 x0 . Next, we consider Q2,1,2 i−1 the graph 2,1 1 obtained from Qi−1 by expanding x0 to x0 vw in such a way that the partition associated with the graph the vertex w is {x1 , x23 }. Then we add the new edge vu. Then, we consider Q2,1,2,2 i−1 1 2 obtained from Q2,1,2 by expanding u to u zu in such a way that the partition associated i−1 with the vertex u2 is {x23 , x1 }. Then we add the new edge zw. We now consider Q2,1,2,2,2,1 i−1 1 ′ 2 the graph obtained from Q2,1,2,2 by expanding w to w v w in such a way that the partition i−1 associated with the vertex w2 is {v, z}. Then we add the new edges u2 v ′ and x2 v ′ . The graph Q2,1,2,2,2,1 contains the desired instance (see Figure 15(c)). i−1 We now show the generation of the configuration depicted in Figure 15(a). Again we split this case into two subcases: (*) at least one vertex in {x1 , x2 , y1 , y2 } has degree 4 in Qi−1 and (**) all vertices in {x1 , x2 , y1 , y2 } have degree 3 in Qi−1 . subcase (*): without loss of generality we suppose that x1 has degree 4 in Qi−1 . We now consider 2 1 Q2,1 i−1 the graph obtained from Qi−1 by expanding x1 to x1 ux1 in such a way that the partition 2 associated with the vertex x1 is {x2 , y1 }. Then we add the new edges uy2 and x11 x2 . Let 2,1 1 ′ 2 Q2,1,2 i−1 be the graph obtained from Qi−1 by expanding x2 to x2 u x2 in such a way that the 2 2 partition associated with the vertex x2 is {x1 , y2 }. Then we add the new edge uu′ . We shall 1 2 consider Q2,1,2,2 obtained from Q2,1,2 i−1 i−1 by expanding y2 to y2 vy2 in such a way that the partition 2 2 associated with the vertex y2 is {u, x2 }. Then we add the new edge vx21 . Let Q2,1,2,2,2,1 be the i−1 2,1,2,2 1 ′ 2 graph obtained from Qi−1 expanding u to u v u in such a way that the partition associated with the vertex u2 is {u′ , y22 }. Then we add the new edges v ′ v and u1 u′ . The graph Q2,1,2,2,2,1 i−1 contains the desired instance (see Figure 15(a)). subcase (**): by Proposition 12 we have that all vertices in {x0 , x3 , y0 , y3 } have degree 4 in Qi−1 . We consider Q2i−1 the graph obtained from Qi−1 by expanding x0 to x10 ux20 in such a way that the partition associated with the vertex x20 is {x1 , x3 }. Then we add the new edges ux2 and 2,1 1 ′ 2 x10 x3 . We consider Q2,1,2 i−1 the graph obtained from Qi−1 by expanding x3 to x3 u x3 in such a 2 2 way that the partition associated with the vertex x3 is {x0 , x2 }. Then we add the new edge 1 2 uu′ . Let Q2,1,2,2 be the graph obtained from Q2,1,2 i−1 i−1 by expanding u to u vu in such a way 2 2 that the partition associated with the vertex u is {x0 , x2 }. Then we add the new edge vx23 . We consider Q2,1,2,2,2,1 the graph obtained from Q2,1,2,2 by expanding x2 to x12 v ′ x22 in such a i−1 i−1 2 way that the partition associated with the vertex x2 is {u2 , x1 }. Then we add the new edge vv ′ and x1 x12 . The graph Q2,1,2,2,2,1 contains the desired instance. i−1 19 5 Concluding remarks The aim of this paper is to provide a not-straightforward inductive understanding of the cubic bridgeless graphs. We show how to inductively embed cubic bridgeless graphs into bigger class S of braces obtained from the ladder by simple augmentations. Next step is to generalize theorems and conjectures on cubic bridgeless graphs as statements about elements of S. This is at present our work in progress. In particular, we have used hexagon graphs for study of dcdc. The paper Cubic bridgeless graphs and braces is first one in the series of three papers ([2, 3]), where the aim has been to confirm an intuition that ”in a sense” all the partial obstacles to validity of dcdc are of cut-type. We managed to confirm this intuition in the last paper ([3]). In this series of papers, we use the machinery of hexagon graphs; not the simple augmentations in a fundamental way though. Nevertheless, the hexagon graphs environment was essential to get into the complex structure of partial obstacles. As mentioned above, we believe that other conjectures on cubic bridgeless graphs can be formulated for class S, even though at present we do not have a reformulation even for dcdc. References [1] F. Jaeger. A survey of the cycle double cover conjecture. In B.R. Alspach and C.D. Godsil, editors, Annals of Discrete Mathematics 27 Cycles in Graphs, volume 115 of North-Holland Mathematics Studies, pages 1 – 12. North-Holland, 1985. [2] A. Jiménez and M. Loebl. Directed cycle double covers: Fork graphs. arXiv:1310.5539, 2013. [3] A. Jiménez and M. Loebl. Directed cycle double covers and cut-obstacles. arXiv:1405.6929, 2014. [4] R. Kenyon. The laplacian and dirac operators on critical planar graphs. Inventiones mathematicae, 150(2):409–439, 2002. [5] L. Lovász. Matching structure and the matching lattice. Journal of Combinatorial Theory, Series B, 43(2):187 – 222, 1987. [6] L. Lovász and M.D. Plummer. Matching Theory. Akadémiai Kiadó, Budapest, 1986. Also published as Vol. 121 of the North-Holland Mathematics Studies, North-Holland Publishing, Amsterdam. [7] W. McCuaig. Brace generation. Journal of Graph Theory, 38(3):124–169, 2001. [8] Ch. Mercat. Discrete riemann surfaces and the ising model. Communications in Mathematical Physics, 218(1):177–216, 2001. [9] B. Mohar and C. Thomassen. Graphs on surfaces. Johns Hopkins studies in the mathematical sciences. Johns Hopkins University Press, Baltimore (MD), London, 2001. [10] C.-Q. Zhang. Integer flows and cycle covers of graphs. Marcel Dekker, Inc, 1997. 20