Every 5-connected planar triangulation is 4-ordered Hamiltonian

J. Algebra Comb. Discrete Appl. 2(2) • 111-116 Received: 24 December 2014; Accepted: 14 March 2015 DOI 10.13069/jacodesmath.42463 Journal of Algebra Combinatorics Discrete Structures and Applications Every 5-connected planar triangulation is 4-ordered Hamiltonian∗ Research Article Kenta Ozeki∗∗ National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan and JST, ERATO, Kawarabayashi Large Graph Project Abstract: A graph G is said to be 4-ordered if for any ordered set of four distinct vertices of G, there exists a cycle in G that contains all of the four vertices in the designated order. Furthermore, if we can find such a cycle as a Hamiltonian cycle, G is said to be 4-ordered Hamiltonian. It was shown that every 4-connected planar triangulation is (i) Hamiltonian (by Whitney) and (ii) 4-ordered (by Goddard). Therefore, it is natural to ask whether every 4-connected planar triangulation is 4-ordered Hamiltonian. In this paper, we give a partial solution to the problem, by showing that every 5connected planar triangulation is 4-ordered Hamiltonian. 2010 MSC: 05C10 Keywords: 4-ordered, 4-ordered Hamiltonian, Triangulations, Plane graphs 1. Introduction A graph G is said to be k-ordered for an integer 3 ≤ k ≤ |V (G)|, if for any ordered set of k distinct vertices of G, there exists a cycle in G that contains all the k vertices in the designated order. Furthermore, if we can find such a cycle as a Hamiltonian cycle, G is said to be k-ordered Hamiltonian. These topics have been extensively studied; see the survey [2]. In this paper, we focus on 4-connected planar triangulations. In fact, it is known that such graphs have good properties; Theorem 1.1. Let G be a 4-connected planar triangulation. Then (i) G is Hamiltonian. (Whitney [12]) ∗ This work was in part supported by JSPS KAKENHI Grant Number 25871053 and by Grant for Basic Science Research Projects from The Sumitomo Foundation. ∗∗ E-mail: [email protected] JACODESMATH / ISSN 2148-838X 111 5-connected planar triangulation (ii) G is 4-ordered. (Goddard [3]) Note that Theorem 1.1 (i) was improved to 4-connected planar graphs (by Tutte [11]) and 4-connected projective planar graphs (by Thomas and Yu [8]). However, we cannot lower the assumption on 4connectedness to 3-connectedness, since there exist infinitely many 3-connected planar triangulations that are not Hamiltonian (see [4]). On the other hand, by using ideas of Goddard [3], we can construct infinitely many 3-connected planar triangulations that are not 4-ordered, and infinitely many 5-connected planar graphs that are not 4-ordered. Therefore, both of the assumptions “4-connected” and “triangulation” are needed for the property of being “4-ordered”. Recall that 4-ordered Hamiltonian graphs are definitely Hamiltonian and 4-ordered. It follows from Theorems 1.1 (i) and (ii) that every 4-connected planar triangulation satisfies both properties, and hence it is natural to pose the following conjecture (Conjecture 1.2). The main purpose of this paper is to show Theorem 1.3, which is a partial solution to it. Conjecture 1.2. Every 4-connected planar triangulation is 4-ordered Hamiltonian. Theorem 1.3. Every 5-connected planar triangulation is 4-ordered Hamiltonian. This paper is organized as follows; in the next section, we will give terminologies and a known result, used in the proof of Theorem 1.3 in Section 3. In the last section, we will give a conclusion of this paper, together with some open problems. 2. Preliminaries For a graph G, the order of G is denoted by |G|. Let H
1 and H2 be two subgraphs of a graph
G. Then H1 ∪ H1 denotes the subgraph of G with V H1 ∪ H2 = V (H1 ) ∪ V
(H2 ) and E H1 ∪ H2 = E(H1 ) ∪ E(H
2 ), and H1 ∩ H1 denotes the subgraph of G with V H1 ∩ H2 = V (H1 ) ∩ V (H2 ) and E H1 ∩ H2 = E(H1 ) ∩ E(H2 ). We use a similar notation
also for a vertex subset U or
an edge subset P of G; so, H1 ∪ U is the subgraph of G with
V H1 ∪ U = V (H1 ) ∪ U and E H
1 ∪ U = E(H1 ), and H1 ∪ P is the subgraph of G with V H1 ∪ P = V (H1 ) ∪ V (P ) and E H1 ∪ P = E(H1 ) ∪ P , where V (P ) is the set of vertices that are end vertices of some edges in P . A pair (H1 , H2 ) is a separation of G if H1 ∪ H2 = G and E(H1 ) ∩ E(H2 ) = ∅. For a path P and two vertices x, y ∈ V (P ), P [x, y] denotes the subpath of P between x and y. Furthermore, let P (x, y] = P [x, y] − {x}, P [x, y) = P [x, y] − {y}, and P (x, y) = P [x, y] − {x, y}. Let G be a connected plane graph. A facial walk in G is the boundary walk of some face of G. Furthermore, if it is a cycle, then we call it a facial cycle in G. Let T be a subgraph of a graph G. A T -bridge of G is either (i) an edge of G − E(T ) with both ends on T or (ii) a subgraph of G induced by the edges in a component of G − V (T ) and all edges from that component to T . A T -bridge satisfying (i) is said to be trivial ; otherwise it is non-trivial. For a T -bridge B of G, the vertices in B ∩ T are the attachments of B (on T ), and any vertex of B that is not an attachment is a non-attachment. We say that T is a Tutte subgraph in G if every T -bridge of G has at most three attachments on T . For another subgraph C of G, T is a C-Tutte subgraph in G if T is a Tutte subgraph in G and every T -bridge of G containing an edge of C has at most two attachments on T . When T is a path or a cycle, we call T a C-Tutte path or a C-Tutte cycle, respectively. Note that if G is 4-connected and T is a Tutte subgraph in G with |T | ≥ 4, then T must contain all vertices in G; otherwise, there exists a T -bridge in G whose attachments form a cut set in G of order at most three, contradicting that G is 4-connected. Indeed, the concept of “Tutte subgraphs” was first introduced by Tutte [11] in order to prove his seminal result; every 4-connected planar graph is Hamiltonian. Since then it has been extended by several researchers, see [6–9, 13]. The following theorem is a main tool to prove Theorem 1.3. See also the paper [9] by Thomassen. 112 K. Ozeki Theorem 2.1 (Sanders [7]). Let G be a connected plane graph, let C be a facial walk in G, let x, y ∈ V (G) with x 6= y, and let e ∈ E(C). Assume that G contains a path from x to y through e. Then G has a C-Tutte path from x to y through e. Note that originally Sanders [7] showed only the 2-connected case, but we can easily show Theorem 2.1 using a block decomposition. Hence, we omit the proof of Theorem 2.1. 3. Proof of Theorem 1.3 Let G be a 5-connected planar triangulation, and let v1 , v2 , v3 and v4 be four distinct vertices in G. We will show that G has a Hamiltonian cycle passing through v1 , v2 , v3 and v4 in this order. It follows from Theorem 1.1 (ii) that G has a cycle passing through those vertices in that order. This implies the following; G − v4 has a path P from v1 to v3 through v2 such that
(P1) G − V P (v1 , v3 ) contains a path from v3 to v1 through v4 . In addition, by taking a path satisfying property (P1) as short as possible, we can also consider the following condition. Here a chord of P is an edge e not in P such that both of end vertices of e are contained in P . (P2) For any chord of P , one end vertex of it is contained in P [v1 , v2 ) and the other is contained in P (v2 , v3 ]. Indeed, if there exists a chord xy of P such that both end vertices are contained in P [v1 , v2 ] or in P [v2 , v3 ], then we can detour P by xy instead of P [x, y]. It is easy to see that the new path also satisfies condition (P1) and is shorter than P . Therefore, a path that is as short as possible, subject to (P1), also satisfies condition (P2).
Let G1 = G − V P (v1 , v3 ) , and let C 1 be the unique facial walk of G1 that is not facial in G. Note that v1 , v3 ∈ V (C 1 ). Now we consider a separation (H1 , H2 ) of G1 such that |H1 ∩ H2 | ≤ 2, v1 , v3 ∈ V (H1 ), and v4 ∈ V (H2 ) − V (H1 ). When H1′ consists of only the two vertices v1 and v3 and no edges and H2′ = G1 , a pair (H1′ , H2′ ) is a separation of G1 satisfying all of the above conditions. Therefore, such a separation (H1 , H2 ) of G1 must exist. Take such a separation (H1 , H2 ) of G1 so that |H2 | is as small as possible. If H2 does not contain any edge in C 1 , then H1 ∩ H2 forms a cut set in G of order at most 2, contradicting that G is 5-connected. Hence H2 contains an edge in C 1 . Then it follows from condition (P1) that |H1 ∩ H2 | = 2 and there exists an edge e1 in H2 ∩ C 1 such that G1 has a path from v3 to v1 through e1 . (In fact, take an edge in H2 ∩ C 1 such that it is incident with a vertex in H1 ∩ H2 .) It follows from Theorem 2.1 that G1 has a C 1 -Tutte path T 1 from v3 to v1 through e1 . Note that by the choice of e1 , T 1 passes through both of the two vertices in H1 ∩ H2 . In addition, it satisfies the following property. Claim 3.1. T 1 contains v4 , but does not contain v2 . Proof. Since v2 6∈ V (G1 ), the second statement is trivial. So we only show the first one. Suppose not, and let B be a T 1 -bridge of G1 such that B contains v4 as a non-attachment. Let SB be the set of attachments of B on T 1 . If B has no neighbors in P (v1 , v3 ), then SB would be a cut set of G such that SB separates B − SB from other vertices and |SB | ≤ 3, which contradicts that G is 5-connected. Therefore, B has neighbors in P (v1 , v3 ). This implies that B contains
an edge in C 1 . Then since T 1 is a C 1 -Tutte path in G1 , we have |SB | ≤ 2. Let B = G1 − V B − SB . Then (B, B) is a separation of G1 such that |B ∩ B| = |SB | ≤ 2, v1 , v3 ∈ V (B) and v4 ∈ V (B) − V (B). Furthermore, since T 1 passes through e1 and B is a T 1 -bridge of G1 , we have e1 ∈ E(B), which implies that V (H2 ) − V (B) 6= ∅. Since T 1 passes through both of the two vertices in H1 ∩ H2 , we see that V (B) ⊂ V (H2 ), which contradicts the choice of (H1 , H2 ). This completes the proof of Claim 3.1. 113 5-connected planar triangulation
Let G2 = G − V T 1 (v3 , v1 ) , and let C 2 be the unique facial walk of G2 that is not facial in G. Note that v1 , v3 ∈ V (C 2 ). Then we consider a separation (R1 , R2 ) of G2 such that |R1 ∩ R2 | ≤ 2, v1 , v3 ∈ V (R1 ), and v2 ∈ V (R2 ) − V (R1 ). When R1′ consists of only the two vertices v1 and v3 and no edges and R2′ = G2 , a pair (R1′ , R2′ ) is a separation of G2 satisfying all of the above conditions. Therefore, such a separation (R1 , R2 ) of G2 must exist. Take such a separation (R1 , R2 ) so that |R2 | is as small as possible. Since G is 5-connected, R2 contains an edge in C 2 . Note that P is contained in G2 , and hence G2 has a path from v1 to v3 through v2 . This implies |R1 ∩ R2 | = 2 and there exists an edge e2 in R2 ∩ C 2 such that G2 has a path from v1 to v3 through e2 . It follows from Theorem 2.1 that G2 has a C 2 -Tutte path T 2 from v1 to v3 through e2 . Note that by the choice of e2 , T 2 passes through both of the two vertices in R1 ∩ R2 . Notice also that T 1 ∪ T 2 is a cycle in G, and it satisfies the following, which is crucial in the proof of Theorem 1.3.
Claim 3.2. There exist no non-trivial T 1 ∪ T 2 -bridges in G. In particular, T 1 ∪ T 2 is a Hamiltonian cycle in G.
Proof. Suppose that there exists a non-trivial T 1 ∪ T 2 -bridge D in G. Let SD be the set of attachments of D on T 1 ∪ T 2 .
Suppose first that SD ∩ V T 1 (v3 , v1 ) = ∅. This condition implies that D is a T 2 -bridge of G2 . Since T 2 is a C 2 -Tutte path in G2 , we have |SD | ≤ 3, which implies that SD is a cut set in G of order
at most 1 three, contradicting that G is 5-connected. Therefore, we may assume that S ∩ V T (v , v ) 6= ∅. By D 3 1
the same argument, we also see that SD ∩ V T 2 (v1 , v3 ) 6= ∅. These conditions, together with imply that D contains an edge in C 1 and an edge in
the planarity, 1 2 C . Then since D − V T (v3 , v1 ) is a T -bridge of G2 containing an edge in C 2 , we have 2 |SD ∩ V (T 2 )| ≤ 2. (1) Suppose that D contains
no vertices in P as non-attachments. See Figure 1. This condition implies
that there exists a T 1 ∪ P -bridge, say BD , such that D ⊆ BD . Note that BD − V
P (v1 , v3 ) is connected and a T 1 -bridge of G1 containing an edge in C 1 , and
hence BD − V P (v1 , v3 ) has at most
1 1 two attachments on T . Since any vertex in SD ∩ V T (v3 , v1 ) is an attachment of BD − V P (v1 , v3 ) on T 1 , we have 

BD − V P (v1 , v3 ) ∩ V (T 1 ) ≤ 2. |SD ∩ V T 1 (v3 , v1 ) | ≤ Then it follows from inequality (1) that |SD | ≤ 4, which contradicts that G is 5-connected. Therefore, we may assume that D contains vertices in P as non-attachments. See Figure 2. Since P is a path in G2 from v1 to v3 and v1 , v3 ∈ V (T 2 ), D has at least two attachments on P . Then it follows from inequality (1) that SD ∩V (T 2 ) ⊆ V (P ) and |SD ∩V (T 2 )| = 2. Let {x, y} = SD ∩V (T 2 ). Note that P [x, y] is contained in D. Consider the region bounded by P [x, y] ∪ T 2 [x, y]. Since SD ∩ V (T 2 ) = SD ∩ V (P ) is the set of attachments of D on T 2 , there are no edges between vertices in P (x, y) and those in T 2 (x, y). Thus, since G is a triangulation, there exists an edge in G connecting x and y. Note that xy is a chord of P . It follows from condition (P2) and symmetry that we may assume that x is contained in P
[v1 , v2 ) and y is contained in P (v2 , v3 ]. Then (D, D) is a separation of G2 , where D = G2 − V D − SD , such that |D ∩ D| = and v2 ∈ V (D). Furthermore, since T 2 passes through e2 and D
|SD | = 2, v1 , v3 ∈ V (D), 1 2 2 is a T ∪ T -bridge of G, we have e ∈ E(D), which implies that V (R2 ) − V (D) 6= ∅. Since T 2 passes through both of the two vertices in R1 ∩ R2 , we see that V (D) ⊂ V (R2 ), which contradicts the choice of (R1 , R2 ).
Therefore, there exist no non-trivial T 1 ∪ T 2 -bridges in G, which easily implies that T 1 ∪ T 2 is a Hamiltonian cycle in G. This completes the proof of Claim 3.2. By Claim 3.1, v2 appears in T 2 and v4 appears in T 1 , which implies that T 1 ∪ T 2 contains v1 , v2 , v3 and v4 in this order. By Claim 3.2, T 1 ∪ T 2 is a Hamiltonian cycle in G. These complete the proof of Theorem 1.3. 114 K. Ozeki D T1 BD P T2 Figure 1. The case when D contains no vertices in P as non-attachments. T1 D v2 x Figure 2. 4. y P T2 The case when D contains vertices in P as non-attachments. Conclusion In this paper, we have focused on the property of being 4-ordered Hamiltonian. In fact, considering known results (Theorem 1.1) on 4-connected planar triangulations, it is natural to pose Conjecture 1.2. We gave a partial solution to it, by showing that every 5-connected planar triangulation is 4-ordered Hamiltonian. In the rest, we would like to put some problems related to k-ordered Hamiltonian. The first one is Conjecture 1.2, which already appeared in Section 1. The second problem is the property of being 4-ordered Hamiltonian of graphs on non-spherical surfaces. In fact, there are some results that are the counterparts of Theorem 1.1. Recall that for a graph G on a non-spherical surface F 2 , the edge-width of G is the length of a shortest non-contractible cycle in G. Theorem 4.1 (Mukae and Ozeki [5]). Let G be a 4-connected triangulation on a surface. Then G is 4-ordered. Theorem 4.2 (Yu [13]). For any surface F 2 , there exists an integer N = N (F 2 ) satisfying the following; for any 5-connected triangulation G of F 2 , if the edge-width of G is at least N , then G is Hamiltonian. Note that the assumptions on 5-connectedness and edge-width in Theorem 4.2 are both best possible, in some sense. In fact, Theorem 4.2 cannot be improved to 4-connected graphs (see [10]) and to the statement without the edge-width assumption (see [1]). Considering these two theorems, the following seems also a natural conjecture. Because of the facts mentioned above, the assumptions on the edge-width and 5-connectedness are best possible, if the conjecture is true. We leave it to readers as an open problem. Problem 4.3. For any surface F 2 , there exists an integer N = N (F 2 ) satisfying the following; for any 5-connected triangulation G of F 2 , if the edge-width of G is at least N , then G is 4-ordered Hamiltonian. Goddard [3] also mentioned about the property of being 5-ordered; no planar graph can be 5-ordered. However, his idea cannot work for graphs on non-spherical surfaces, and hence the following might also hold. Those are the last problems in this paper. 115 5-connected planar triangulation v1 v4 Figure 3. v3 Two adjacent vertices of degree 4. Problem 4.4. Any 5-connected triangulation of a non-spherical surface F 2 is 5-ordered. Problem 4.5. For any surface F 2 , there exists an integer N = N (F 2 ) satisfying the following; for any 5-connected triangulation G of F 2 , if the edge-width of G is at least N , then G is 5-ordered Hamiltonian. Note that if a 4-connected triangulation G of a surface has two adjacent vertices of degree 4, then G cannot be 5-ordered. In fact, if v1 , v3 and v4 are specified as in Figure 3 and v2 and v5 are specified as vertices outside of the structure, then the graph cannot have a cycle containing v1 , v2 , v3 , v4 and v5 in this order. Hence the assumption on 5-connectedness in Problems 4.4 and 4.5 are best possible, in a sense. References [1] D. Archdeacon, N. Hartsfield, C. H. C. Little, Nonhamiltonian triangulations with large connectivity and representativity, J. Combin. Theory Ser. B, 68, 45-55, 1996. [2] R. J. Faudree, Survey of results on k-ordered graphs, Discrete Math., 229, 73-87, 2001. [3] W. Goddard, 4-connected maximal planar graphs are 4-ordered, Discrete Math., 257, 405-410, 2002. [4] J. W. Moon and L. Moser, Simple paths on polyhedra, Pacific J. Math., 13, 629-631, 1963. [5] R. Mukae and K. Ozeki, 4-connected triangulations and 4-orderedness, Discrete Math.,310, 22712272, 2010. [6] K. Kawarabayashi and K. Ozeki, 4-connected projective planar graphs are hamiltonian-connected, (to appear in) J. Combin. Theory Ser. B. [7] D. P. Sanders, On paths in planar graphs, J. Graph Theory, 24, 341-345, 1997. [8] R. Thomas and X. Yu, 4-connected projective-planar graphs are Hamiltonian, J. Combin. Theory Ser. B, 62, 114-132, 1994. [9] C. Thomassen, A theorem on paths in planar graphs, J. Graph Theory, 7, 169-176, 1983. [10] C. Thomassen, Trees in triangulations, J. Combin. Theory Ser. B, 60, 58-62, 1994. [11] W. T. Tutte, A theorem on planar graphs, Trans. Amer. Math. Soc., 82, 99-116, 1956. [12] H. Whitney, A theorem on graphs, Ann. of Math., 32, 378-390, 1931. [13] X. Yu, Disjoint paths, planarizing cycles, and spanning walks, Trans. Amer. Math. Soc., 349, 13331358, 1997. 116