Hypermetric Graphs

arXiv:math/9909185v1 [math.CO] 30 Sep 1999 On equicut graphs Michel Deza∗ Dmitrii V. Pasechnik† September 9, 2018 Abstract The size sz(Γ) of an ℓ1 -graph Γ = (V, E) is the minimum of nf /tf over all the possible ℓ1 -embeddings f into nf -dimensional hypercube with scale tf . The sum of distances between all the pairs of vertices of Γ is at most sz(Γ)⌈v/2⌉⌊v/2⌋ (v = |V |). The latter is an equality if and only if Γ is equicut graph,Pthat is, Γ admits an ℓ1 -embedding f that for any 1 ≤ i ≤ nf satisfies x∈X f (x)i ∈ {⌈v/2⌉, ⌊v/2⌋} for any x ∈ V . Basic properties of equicut graphs are investigated. A construction of equicut graphs from ℓ1 -graphs via a natural doubling construction is given. It generalizes several well-known constructions of polytopes and distanceregular graphs. Finally, large families of examples, mostly related to polytopes and distance-regular graphs, are presented. 1 Background A finite simple graph Γ = (V, E) with v = v(Γ) vertices has the natural graph distance denoted by d(x, y) = dΓ (x, y). The diameter of Γ, that is, that maximum of d(x, y) over all the pairs of vertices x, y, is denoted by D(Γ). Then Γ = (V, E) is an ℓ1 -graph if there exists a mapping f : V 7→ Rm with dΓ (x, y) being equal to the ℓ1 -distance between f (x) and f (y) for any two x, y ∈ V or, equivalently, if there exists a mapping g : V 7→ H n (the latter being the vertexset of the n-cube) such that for any x, y ∈ V we have td(x, y) = dH (g(x), g(y)) for some integer t = tf . Recall that dH (g(x), g(y)) is the Hamming distance, that is, the size of the symmetric difference of the sets with characteristic vectors g(x) and g(y). The number t is called the scale of the embedding f . For more information on ℓ1 -graphs we refer the reader to [DeLa97, Chapt. 21]. In fact, such t should be 1 or even number that is at most v − 2 (assuming v ≥ 4). Call the smallest such t = tf (if any) the scale t(Γ) of Γ. Call the the minimum of n/t the size sz(Γ) of ℓ1 -graph Γ. (In general, sz(Γ) is larger than minimal m, the dimension of the host ℓ1 -space into which the shortest path metric of Γ embeds isometrically.) ∗ CNRS † Dept. and Ecole Normale Supérieure, Paris, France of Computer Science, Utrecht University, The Netherlands 1 W (Γ) denotes the sum of all v(v − 1)/2 pairwise distances between vertices of Γ; chemists call it the Wiener number. In [Ple84] this number is called transmission. A powerful tool to deal with ℓ1 -graphs is via linear programming. Namely, consider the cone M et(V ) of semimetrics on V . The elements of M et(V ) are symmetric v × v-matrices with 0 diagonal that satisfy the triangle inequalities. A metric is a semimetric that does not have 0’s outside the main diagonal. The distance matrix Dist(Γ) of a graph Γ is a point in M et(V ). The set of cuts S of V consists of partitions of V into two parts. Each s ∈ S defines an extremal ray of the cone M et(V ). Namely, s defines a semimetric δs on V , so that δs (u, v) = 0 or 1 depending on u, v ∈ V being in same or different parts of s. Then one may ask whether Dist(Γ) lies in the cone Cut(V ) that is generated by the cuts, that is X Dist(Γ) = λs δs (1) s∈S for the appropriate choice of λs . Hence Dist(Γ) ∈ Cut(V ) implies that X t · Dist(Γ) = λs δs (2) s∈S for an integer t and an integer vector λ = (λs | s ∈ S). The above gives another definition of the ℓ1 -graphs: a graph Γ is ℓ1 -graph if and only if Dist(Γ) ∈ Cut(V ). Each equality (2) for a given Pt and λ defines an t-scale embedding f of Γ into the hypercube of dimension s∈S λs . An advantage of using (2) is that is allows to classify ℓ1 -embeddings of Γ up to equivalence: different solutions to (2) with nonnegative integer unknowns λ such that g.c.d.(t, λi ) = 1 correspond to different embeddings. If such a solution is unique Γ is called rigid. Any rigid ℓ1 -graph has scale 1 or 2; any ℓ1 -graph of scale 1 is rigid. If an ℓ1 -graph does not contain K4 then it is rigid. Cf. [DeLa97, Sect. 21.4] for further information. It is easy to check the following reformulation of [DeLa97, (4.3.6)]. Lemma 1.1 W (Γ)/(⌈v/2⌉⌊v/2⌋) ≤ sz(Γ) ≤ W (Γ)/(v − 1). Proof. Let the (t-scale) embedding f be given by the matrix F = (Fiu ) = (f (u)i ). The columns of F can be regarded as characteristic vectors of one of the two parts of cuts s ∈ S. Using (2) and summing up the dΓ (u, v)’s, we obtain X X X λs δs (u, v) W = W (Γ) = dΓ (u, v) = u,v∈V s∈S u,v∈V = X X λs δs (u, v) = s∈S u,v∈V ≤ ⌈v/2⌉⌊v/2⌋ X s∈S X λs , s∈S 2 λs |s|(v − |s|) by taking the maximum over |s| in (3). P On the other hand, by taking the ✷ minimum there, we obtain W ≥ (v − 1) s∈S λs . When the equality holds in the left-hand side of the inequality in Lemma 1.1, we say that Γ is an equicut graph. This means that for such a graph in the formulae (2) every s ∈ S satisfies δs 6= 0 if and only if s partitions V into parts of size ⌈v/2⌉ and ⌊v/2⌋. On the other hand, the equality on the other side only happens in very special case. Lemma 1.2 sz(Γ) = W (Γ)/(v − 1) if and only if Γ is the star K1,v−1 . Proof. Without loss in generality, the first row f (u) of the matrix F defined by f consists of 0’s. Moreover, all the cuts s that appear in F are partitions of V into part of size one and of size v − 1. It follows that the any geodesic between x, y ∈ V must contain u. Hence Γ is a star. ✷ Further, the size sz(Γ) has the following properties formulated in terms of v. Theorem 1.3 Let Γ be an ℓ1 -graph. (i) sz(Γ) = 2 − 1/(⌈v/2⌉) if and only if Γ = Kv ; (ii) sz(Γ) = 2 if and only if Γ is a non-complete subgraph of a Cocktail-Party graph; (iii) sz(Γ) = v − 1 if and only if Γ is a tree; (iv) 2 < sz(Γ) < v − 1 otherwise. Proof. By taking all the equicuts on V , one obtains a realization of Kv with n/t = 2 − 1/(⌈v/2⌉). By using the left-hand side of the inequality in Lemma 1.1 and the fact that all distances in Kv are 1, we obtain that n/t = sz(Γ), as required in (i). The size of the Cocktail Party graph is 2, as it is easy to construct such a realization (see for instance, Theorem 3.4 below). Then observe that sz(Γ) < 2 implies that Γ is complete, as a two-path from x to y the vertex x is realized, without loss of generality, by the zero vector, and the vertex y is realized by a vector with 2t ones. Hence any non-complete subgraph subgraph of the Cocktail Party graph has size 2, and (ii) is proved. To prove the last two claims, consider the decomposition (1). Choose an s∗ ∈ S with λs∗ > 0; it defines the partition V + (s∗ ) ∪ V − (s∗ ) = V . It follows that the subgraph of Γ induced on V + = V + (s∗ ) is isometric. Indeed, consider a shortest path between x+ and y + . Let k ∈ V lie on this path. Then dΓ (x+ , k) + = dΓ (k, y + ) − dΓ (x+ , y + ) = X λs (δs (x+ , k) + δs (k, y + ) − δs (x+ , y + )) = 0. s∈S 3 As λ ≥ 0, we have δs∗ (x+ , k) + δs∗ (k, y + ) = δs∗ (x+ , y + ). By definition of cut, right-hand side of the above is 0. Hence both the summands in left-hand side are 0, and k ∈ V + . as required. Similarly, the subgraph induced on V − (s∗ ) is isometric. For an edge (x, y) ∈ V + × V − we have dΓ (x, y) = 1 ≥ λs . (Such an edge exists, as Γ is connected.) Hence λ ≤ 1. Let S ⊆ S denote the set of s ∈ S satisfying λs > 0. Now we proceed by induction on v. Let s′ ∈ S. Hence the subgraph ∆+ of Γ induced on V + (s′ ) is isometric. Thus X Dist(∆+ ) = λs δV + (s)∩V + (s′ ) . s∈S−{s′ },V + (s)∩V + (s′ )6=∅ By the induction assumption, X + := X λs ≤ |V + (s′ )| − 1. X λs ≤ |V − (s′ )| − 1. s∈S−{s′ },V + (s)∩V + (s′ )6=∅ Similarly, we obtain X − := s∈S−{s′ },V + (s)∩V − (s′ )6=∅ Hence sz(Γ) = X λs = λs′ + X + + X − ≤ 1 + |V + (s′ )| − 1 + |V − (s′ )| − 1 = (3) s∈§ = v − 1. Finally, if sz(Γ) = v − 1 then (3) turns into an equality, and λs′ = 1. The latter and the induction implies that Γ is a tree. ✷ For a tree Γ, one can moreover show the following. Lemma 1.4 W (K1,v−1 ) = (v − 1)2 , W (Pv ) = v(v − 1)(v + 1)/6. For any tree Γ that is neither the star K1,v−1 nor the path Pv , the value W (Γ) lies strictly within the extremities above. 2 Generalities on equicut graphs Recall that a connected graph Γ = (V, E) is called 2-connected (or 2-vertexconnected) if it remains connected after deletion of any vertex. Lemma 2.1 An equicut graph Γ with a least 4 vertices is 2-connected. Proof. Assume that Γ is not 2-connected. Then there exists a vertex x ∈ V so that the subgraph induced on V − {x} has two connected components, V1 and V2 . We can assume, without loss of generality, that x is represented by all-0 4 vector F (x) in the equicut realization F of Γ. Then for any vi ∈ Vi (i = 1, 2) one has dΓ (v1 , v2 ) = dΓ (v1 , x) + dΓ (x, v2 ) = |v1 | + |v2 |, where |vi | denotes the number of 1’s in the vector F (vi ). Hence the vectors F (v1 ) and F (v2 ) have disjoint supports, for any v1 ∈ V1 and v2 ∈ V2 . That is, any column of F either has 1’s in V1 , or in V2 , but never in both of them. As F is an equicut, we obtain |Vi | ≤ 1, as required. ✷ This implies Corollary 2.2 For any equicut graph Γ with v ≥ 4 vertices, we have 2 − 1/(⌈v/2⌉) ≤ sz(Γ) ≤ v/2 with equality on the left if and only if Γ = Kv and on the right if and only if Γ = Cv . Proof. The left hand side of the inequality follows from Lemma 1.3. Then, to see that Kv is equicut, it suffices to observe that the embedding f given by the matrix F with the columns being the cuts with the smallest part of size ⌊v/2⌋ is an equicut embedding of Kv . To see that Cv is equicut, observe that f given by the matrix F obtained by the cyclic shifts of the row of the form (⌈v/2⌉ times 1,⌊v/2⌋ times 0) produces an equicut realization. To show the last claim of the lemma and the right hand inequality, is suffices to use Lemma 2.1 and the result from [Ple84] that the Wiener number W (Γ) of a 2-connected graph on v vertices is maximal for the circuit Cv (see [Ple84, Th. 5]). ✷ The condition v ≥ 4 is necessary in the statements above. Indeed, sz(P3 ) = 2 > sz(C3 ) = 3/2. Note that P2 and P3 are the only equicut trees. Also, W (C5 ) = 15 < W (P123452 ) and sz(C5 ) = 5/2 < sz(P123452 ), where P123452 denotes the circle on 2 . . . 5 with an extra edge attached to the vertex 2. Remark. Γ is equicut graph if there is a realization with the binary matrix F with the column sums ⌈v/2⌉ or ⌊v/2⌋. If, instead of this condition, we asked that any row of F has exactly k 1’s, then we obtain other special ℓ1 -graph. Namely, one which embeds isometrically up to scale t, into the Johnson graph J(v, k). It is observed in [Shp98] that such graphs can be recognized in polynomial time using the algorithm in [Shp93]; the “atom graph” of Γ is bipartite. However, we are not aware of any similar characterization of the equicut graphs. 5 3 Doublings of ℓ1 -graphs Call an equicut ℓ1 -graph an antipodal doubling if its realization (i.e. above (0, 1)
A O ′ ′ matrix) has form F = J−A J ′ , where A is a v/2 × n (0,1)-matrix, J, J are ′ matrices consisting of 1’s only and O is v/2 × (n − n ) matrix consisting of 0’s. If moreover the matrix A is a realization, with the same scale t, of a graph Γ′ , then it is straightforward to check that J ′ has t(D(Γ′ ) + 1) − n columns. Note that Double Odd graph DO2s+1 (see [BCN89]) with s ≥ 3 is an example of antipodal doubling with the matrix A not corresponding to the realization of a graph Γ′ , for any decomposition of F of the above form. Remark. An antipodal doubling is exactly an ℓ1 -graph that admits an antipodal isomorphism, i.e. it has a central symmetry (for any vertex, there is exactly one other on the distance equal to the diameter) and the mapping of all vertices into their antipodes is an isomorphism. Antipodal extensions of arbitrary ℓ1 -metrics was considered in [DeLa97, Sect. 7.2]. In order to investigate when one can construct an ℓ1 -graph from an ℓ1 graph via the antipodal doubling (see Theorem 3.4 below), let us introduce the following definition. For a graph Γ = (V, E), define its diametral doubling as the graph ✷Γ with the vertex set V + ∪ V − (where V ∗ is a copy of V ) and the adjacency as follows: xµ is adjacent to y ǫ if µ = ǫ and (x, y) ∈ E, or if µ 6= ǫ and dΓ (x, y) = D(Γ). Lemma 3.1 The subgraphs of ✷Γ induced on V ∗ are isometric to Γ if and only if dΓ (x, y) ≤ 2 + dΓ (z1 , z2 ) for any x, y, z1 , z2 ∈ V (4) satisfying dΓ (x, z1 ) = dΓ (y, z2 ) = D(Γ). Proof. Clearly, the condition of the lemma is necessary for the subgraphs induced on V ∗ to be geodetic. Indeed, otherwise there can be a shorter path between, say, x+ and y + that passes through a vertex in V − . To prove sufficiency, observe that any geodetic between p+ and q + either lies wholly in V + , or contains a path of the form x+ , z1− , . . . , z2− , y + , with the (sub)path from z1− to z2− lying wholly in V − . In the former case there is nothing to prove. In the latter case, by definition of ✷Γ, one has that dΓ (x, z1 ) = dΓ (y, z2 ) = D(Γ). By the assumption, there is a path of length at most 2 + dΓ (z1 , z2 ) between x and y. Thus the geodetic can be replaced by another geodetic with the path from x+ to y + lying wholly in V + . ✷ Lemma 3.2 Let Γ satisfy the condition of Lemma 3.1. Then d✷Γ (x+ , y − ) = D(Γ) + 1 − dΓ (x, y) if and only if any geodetic in Γ lies on a geodetic of length D(Γ). 6 (5) Proof. We begin by proving that a geodetic (in Γ) between x and y can be extended to a geodetic of length D(Γ). By Lemma 3.1, there is a geodetic from x+ to y − of the form x+ , . . . , z1+ , z2− , . . . , y − such that the path from x+ to z1+ (respectively, from y − to z2− ) lies wholly in V + (respectively, in V − ). We claim that there is a geodetic between z1 and z2 (by definition of ✷Γ they are at distance D(Γ)) that contains x and y. Clearly, d✷Γ (x+ , y − ) = dΓ (x, z1 ) + 1 + dΓ (z2 , y). Using (5), we derive D(Γ) + 1 − dΓ (x, y) = dΓ (x, z1 ) + 1 + dΓ (z2 , y). Hence D(Γ) = (dΓ (z1 , z2 ) =) dΓ (x, z1 ) + dΓ (z2 , y) + dΓ (x, y), as claimed. Now, suppose that a geodetic between x and y can be extended to a geodetic of length D(Γ), say, between z1 and z2 . By reversing the argument above, we derive (5). ✷ Certain properties of Γ are inherited by ✷Γ. Lemma 3.3 Let Γ = (V, E) be a graph satisfying (4) and (5). Then ✷Γ satisfies D(✷Γ) = D(Γ) + 1, (4) and (5). Proof. As (4) and (5) hold for Γ, we have that each p+ ∈ V + has exactly one vertex p at the maximal distance D(✷Γ) = D(Γ) + 1. Now checking (4) is straightforward. To check (5), take a geodetic between x+ and y − . Note that d✷Γ (x+ , y − ) = D(Γ) + 1 − dΓ (x, y). Hence d✷Γ (x+ , y − ) = d✷Γ (y + , y − ) − d✷Γ (x+ , y + ), (6) and the geodetic in question lies on the path from y + to y − , as required. For a geodetic between x+ and y + observe that (6) also proves that it is extendable to the path from y + to y − . ✷ If D(Γ) = 2 then Γ satisfies (4). Moreover, Γ 6= Kv satisfies (5), unless it has an edge (x, y) with Γ(x) = Γ(y). In particular, any strongly regular graph satisfies (4) and (5). Note that ℓ1 -graphs form a rather small sub-family of strongly regular graphs. 7 Switchings. Not always ∆ = ✷Γ is uniquely defining the graph Γ it was constructed from; it can be that ∆ = ✷Γ′ for Γ′ 6∼ = Γ. See Section 4 for many examples of this situation. Γ and Γ′ are related by the following graph operation. The diametral switching of a graph Γ with respect to S ⊂ V (Γ) is a graph Γ′ that is obtained from Γ by retaining the edges that lie within S × S ∪(V − S)× (V − S) and replacing the set of edges from S×(V −S) with the set {(x, y) ∈ S×(V −S) | dΓ (x, y) = D(Γ)}. Note that Seidel switching (see [BCN89]) is an operation that coincides with the diametral switching for graphs of diameter 2. Theorem 3.4 Let Γ = (V, E) be a ℓ1 -graph. Then ✷Γ is an ℓ1 -graph if Γ satisfies (4), (5) and sz(Γ) ≤ D(Γ) + 1, (7) Moreover, if ✷Γ is an ℓ1 -graph then • D(✷Γ) = D(Γ) + 1, ✷Γ satisfies (4), (5) and (7) with equality, and • All the ℓ1 -realizations of ✷Γ are equicut, of the form, up to permutation
A O of rows and columns, and taking complements of columns, J−A J ′ with t(D(Γ) + 1) columns, so that A is an ℓ1 -realization of Γ with scale t. Proof. It is obvious from sz(Γ) ≤ D(Γ) + 1 that there exists a (v × n, say) matrix A that determines an ℓ1 -embedding
of Γ such that n/tA ≤ D(Γ) + 1. A O ′ Let λ = tA (D(Γ) + 1) − n and B = J−A J ′ , where J has λ columns. Identify + the first v rows of B with V and the last v rows of B with V − . By construction, the Hamming distances between rows of B are equal, up to scale t = tA , to the corresponding distances in ✷Γ. Hence ✷Γ is an ℓ1 -graph. In view of Lemma 3.3, (4) and (5) hold for ✷Γ. ✷Γ satisfies (7) with equality, as by construction, sz(✷Γ) = D(Γ) + 1. Taking an ℓ1 -realization B ′ of ✷Γ, we can reorder the rows of B ′ so that, as in B, the first half of the rows corresponds to V + , and the second half corresponds to V − . By reordering the columns and possibly taking their complements, we reduce B ′ to the desired form. The rest of the claim follows from the fact that Γ induced on V + is isometric in ✷Γ. ✷ Remarks. K4 − P3 , K4 − P2 , the Dynkin diagram E6 are examples of ℓ1 graphs satisfying (4), (7), but not (5). Affine Dynkin diagram Ẽ6 An example of a graph that does not satisfy (4) and (7), but does satisfy (5). Note that any Γ with D(Γ) = 2 satisfies (4) and (5). Certainly, not all of them are ℓ1 -graphs, for instance, K2,3 . Also, not all the ℓ1 -graphs of diameter 2 satisfy (7), for instance, K1,4 . In general, for any ℓ1 -graph Γ = (V, E) with |V | ≥ 4 one has D(Γ) ≤ sz(Γ) ≤ D(Γ) + |V | − 3. 8 (8) The equality at the right-hand side of (8) holds if and only if Γ is a star, as can be seen by applying Theorem 1.3 Theorem 3.4 generalizes to arbitrary ℓ1 -graphs Γ the situation for the Cocktail Party graph Kn×2 considered in [DeLa97, Sect. 7.4], see also second part of Lemma 1.3. It implies that the minimal scale of an ℓ1 -embedding of ✷Γ equals the minimal t > 0 so that the metric t · Dist(Γ) embeds isometrically in H t(D(Γ)+1) . In particular case of Γ = K4a Lemma 7.4.6 of [DeLa97] states that such a minimal t ≥ 2a, with the equality holding if and only if there exists a Hadamard matrix of order 4a. 4 Examples In this section we list many examples in no particular order. An equicut ℓ1 -graph Γ is called a doubling if it is an antipodal doubling; moreover, if it is obtained from a graph ∆ as in Theorem 3.4, we give such a representation Γ = ✷∆. All equicut graphs with at most 6 vertices are Cv (2 ≤ v ≤ 6), K4 , K5 , K6 , P3 , 4-wheel and the octahedron. Among those 11 graphs only K4 , K5 , K6 are not rigid and only C4 , C6 and the octahedron are doublings. Cartesian product Γ × Γ′ of two ℓ1 -graphs Γ and Γ′ is ℓ1 -graph if and only if both Γ, Γ′ are (see [DeLa97, Sect. 7.5]); the scale will be least common multiple of their scales and the size will be the sum of their sizes. Moreover, Γ × Γ′ is rigid if and only if they are. Lemma 4.1 If Γ, Γ′ have even number of vertices each, then Γ × Γ′ is an equicut graph if and only if they are. Proof.     B=   F1 ... F1 F2 ... F ...2 ... Fv ... Fv F ′ F′ ... ... F′ ′ ′ Let  F and F be ℓ1 -realizations of Γ and Γ , respectively. Then    , where Fi ’s are rows of F , is an ℓ1 -realization of Γ × Γ′ . Now   let F and F ′ be equicut realizations. It is straightforward that B is an equicut realization when Γ and Γ′ both have even number of vertices. ✷ A Doob graph (the Cartesian product of a number of copies of Shrikhande graph and a number of copies of K4 , see e.g. [BCN89, p.27]) is an example of an equicut graph obtained via Lemma 4.1. It is a (non-rigid and non-doubling) ℓ1 -graph of scale 2. Distance-regular graphs. Here we freely use notation from [BCN89]. Also, a significant use is made of [KoSh94]. The Petersen graph and the Shrikhande graph are both equicut graphs of scale 2 and size 3; both are rigid and are not 9 doublings. The Double Odd graph DO2s+1 is an equicut graph of scale 1 and size 2s + 1. The half-s-cube 21 H(s, 2) is an equicut graph of scale 2 and size s. It is not rigid only for s = 3, 4; it is a doubling if and only if s is even. The Johnson graph J(2s, s) is an non-rigid doubling of scale 2 and size s. Further, the following are distance-regular equicut graphs. 1. All Taylor ℓ1 -graphs - 21 H(6, 2), J(6, 3), C6 , H(3, 2), the Icosahedron. They are all doublings of diameter 3 and size 3 that can be constructed using Theorem 3.4 above. 2. All strongly regular ℓ1 -graphs, except J(s, 2) (s ≥ 5) and grids H(2, s) with s odd. That is, C5 , Petersen, 12 H(5, 2), Shrikhande, H(2, s) with s even, and Ks×2 . 3. Among distance regular graphs Γ with D(Γ) > 2 and µ > 1: 21 H(s, 2) with s > 5, H(s, d), J(s, t) with t > 2, the Icosahedron, and Doob graphs. 4. All Coxeter ℓ1 -graphs except J(s, t) with t < s/2: J(2s, s), the Icosahedron, the Dodecahedron, Ks×2 , 21 H(s, 2), H(s, 2), Cs (s ≥ 5). 5. All cubic distance-regular ℓ1 -graphs: K4 , Petersen, H(3, 2), DO5 , the Dodecahedron. Also all amply regular ℓ1 -graphs with µ > 1 are equicut graphs. Yet another example is given by the 12-vertex co-edge regular subgraph of the Clebsh graph 1 2 H(5, 2), see [BCN89, Sect. 3.11, p. 104]; it is an equicut graph of size 5/2, scale 2, non-doubling. Some equicut graphs which are doublings of ℓ1 -graphs Section 7.2 of [DeLa97]): (see some in 1. C2s = ✷Ps . 2. Ks×2 = ✷Ks . 3. H(s, 2) = ✷H(s − 1, 2). 4. J(2s, s) = ✷J(2s − 1, s). 5. 1 2 H(2s, 2) = ✷ 21 H(2s − 1, 2). 6. P rism2s = ✷C2s = ✷L2s ; here L2s is the “ladder”on 2s vertices. L2s is the diametral switching of C2s with respect to the vertex set of a path Ps ⊂ C2s . 7. AP rism2s+1 = ✷C2s+1 . 8. The Dodecahedron is the doubling of the 9-circle (1, . . . , 9) with extra vertex connected to the vertices 3, 6, 9 of the circle. 10 9. The Icosahedron is the doubling of 5-wheel; as well, it is the doubling of the graph obtained from hexagon (1, . . . , 6) by adding edges (2i, 2j) for 0 < i < j < 4. 10. J(6, 3) is the doubling of Petersen graph, in addition to 4. 11. 1 2 H(6, 2) is the doubling of Shrikhande graph, and the doubling of H(2, 4) (more precisely, of its realization in half-6-cube) in addition to 5. In 9, 10, 11 we have (diametral) switching-equivalent graphs Γ such that ✷Γ is a Taylor ℓ1 -graph; see remark preceding Theorem 3.4. This situation in general is well-known; for instance, the Gosset graph, a Taylor graph that is not an ℓ1 -graph, can be obtained as the diametral doubling of one of 5 non-isomorphic, but switching-equivalent, graphs. For definitions and discussion of this situation in more general setting, see e.g. [BCN89, pp.103-105]. Equicut polytopes. The skeletons of many nice polytopes are equicut graphs. Below we list several such examples. We follow the terminology from [Joh66, Cox73]. All five Platonic solids have equicut skeletons; all, except the Tetrahedron, are rigid. All but the cube (of scale 1), have scale 2. The sizes for the Tetrahedron, the Octahedron, the Cube, the Icosahedron and the Dodecahedron are 3/2, 2, 3, 3 and 5. The skeleton of any zonotope (see e.g. [Zie95] for a definition of zonotope) is a doubling and it has scale 1, so it is rigid. Among the Archimedean ℓ1 -polytopes 1. all zonohedra (i.e. 3-dimensional zonotopes) are as follows: the truncated Octahedron, the truncated Cuboctahedron, the truncated Icosidodecahedron and P rism2s (s > 2) with sizes 6, 9, 15 and s + 1, respectively; 2. all the other doublings are as follows: the Rhombicuboctahedron, the Rhombicosidodecahedron and Aprism2s+1 (s > 1) with scale 2 and sizes 5, 8 and s + 1, respectively; 3. all remaining equicut polytopes - the snub Cube, the snub Dodecahedron and Aprism2s (s > 1) - all have scale 2 and sizes 9/2, 15/2 and s + 1/2, respectively; 4. the remaining, P rism2s+1 (s ≥ 1) has scale 2 and size s + 3/2; it is not an equicut graph. Among all Catalan (dual Archimedean) ℓ1 -polyhedra: 1. all the zonohedra are as follows: dual Cuboctahedron and dual Icosidodecahedron of sizes 4 and 6, respectively; 2. the only other doubling is dual truncated Icosahedron (also known as dual football) of scale 2 and size 5; 11 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 1 0 0 0 1 0 0 0 1 0 0 1 1 0 0 0 1 0 0 0 1 1 0 1 0 0 0 0 1 1 0 0 1 0 1 1 1 0 0 0 1 1 0 0 1 1 0 1 1 0 0 0 1 1 0 1 1 1 0 1 0 0 0 0 1 1 0 1 1 1 1 1 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 1 1 0 0 1 0 0 0 1 0 1 0 1 0 0 0 1 0 0 1 1 0 1 0 0 0 0 0 1 0 1 0 1 0 1 0 1 1 0 0 1 0 1 0 1 1 1 0 1 0 0 0 1 0 1 1 1 0 1 0 1 0 0 0 1 0 1 1 1 1 1 0 1 1 0 0 1 1 1 0 0 0 1 0 1 1 0 0 1 1 1 0 0 1 1 1 1 1 0 0 1 1 1 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 1 1 1 1 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 0 0 0 1 0 0 0 1 1 0 1 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 1 1 1 0 1 0 0 1 0 0 1 1 1 1 0 1 0 0 0 1 0 0 1 1 1 1 1 0 0 0 0 1 0 0 1 1 1 1 1 1 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 1 1 0 1 0 1 0 0 1 1 0 0 0 0 0 1 0 1 0 0 1 1 0 1 1 0 0 1 0 1 0 0 1 1 1 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 0 1 0 1 1 0 1 1 0 1 1 1 0 1 0 1 1 0 1 1 1 1 1 0 0 1 0 1 1 1 1 1 0 1 1 0 0 1 0 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 1 0 1 0 0 0 0 1 1 0 1 1 1 1 1 0 1 0 0 1 1 0 1 1 1 1 1 1 0 0 0 1 1 0 1 1 1 1 1 1 1 1 Figure 1: An equicut realization of the snub 24-cell; only half of the 96 column vectors are shown. The remaining 48 are obtained as complements. 3. all remaining cases (duals of truncated Cube, of truncated Dodecahedron, of truncated Tetrahedron and of P rism3 ) are non-equicut ℓ1 -graphs of scale 2 and of sizes 6, 13, 7/2 and 2, respectively. All Archimedean or Catalan ℓ1 -polyhedra are rigid except the dual P rism3 ; those embeddings are presented in [DeSt96]. Examples of regular-faced polyhedra (from the well-known list of 92 polyhedra, see [Joh66, Za69]) with equicut skeletons, are (all of scale 2) #75 biaugmented P rism6 of size 4 (a doubling) and two non-doublings: #74 augmented P rism6 of size 4 and #83 tridiminished Icosahedron of size 3. The regular ℓ1 -polytopes of dimension greater than 3 have equicut skeletons. They are as follows. • Kv ((v − 1)-simplex); • H(s, 2) (s-cube); • Ks×2 (cross-polytope). There are just 3 semiregular ℓ1 -polytopes of dimension greater than 3, see [DeSh96]. Two of them have equicut skeletons: 12 H(5, 2) and the snub 24-cell s(3, 4, 3), see Figure 1. The latter is a 4-dimensional semiregular polytope with 96 vertices (see, for example, [Cox73]); the regular 4-polytope 600-cell can be obtained by capping its 24 icosahedral facets. Its skeleton has scale 2 and size 6; it is a doubling. Three of the chamfered (see [DeSh96]) Platonic solids have ℓ1 -skeletons: chamfered Cube is a zonohedron of size 7, chamfered Dodecahedron is an equicut (non-doubling) graph of scale 2 and size 11, chamfered Tetrahedron is nonequicut ℓ1 -graph of scale 2 and size 4. References [BCN89] A.E.Brouwer, A.M.Cohen and A.Neumaier, Graphs, Springer-Verlag, Berlin, 1989. 12 Distance-Regular [Cox73] H.S.M.Coxeter, Regular Polytopes, Dover Publications, New York,1973. [DeLa97] M.Deza and M.Laurent, Geometry of cuts and metrics, Algorithms and Combinatorics 15, Springer-Verlag, Berlin, 1997. [DeGr93] M.Deza and V.P.Grishukhin, Hypermetric graphs, Quart. J. of Math. Oxford 2 (1993) 399-433. [DeSh96] M.Deza and S.V.Shpectorov, Recognition of ℓ1 -graphs with complexity O(nm), or football in a hypercube, Eur. J. Comb. 17 (1996) 279-289. [DeSt96] M.Deza and M.I.Shtogrin, Isometric embedding of semi-regular polyhedra, plane partitions and their duals into hypercubes and cubic lattices, Uspechi Math. Nauk 51-6 (1996) 199-200. [Koo90] J. Koolen, On metric properties of regular graphs, Master’s thesis, Technische Universiteit Eindhoven, 1990. [KoSh94] J. Koolen, S.V. Shpectorov, Distance-regular graphs the distance matrix of which has only one positive eigenvalue, Eur. J. Comb. 15 (1994) 269-275. [Ple84] J. Plesnik, On the sum of all the distances in a graph or digraph, J. Graph. Theory 8 (1984) 1-21. [Shp93] S.V. Shpectorov, On scale embedding of graphs into hypercubes, Eur. J. Comb. 14 (1993) 117-130. [Shp98] S.V. Shpectorov, personal communication. [Joh66] N.W. Johnson, Convex polyhedra with regular faces, Canadian J. Math. 18 (1966) 169-200. [Za69] V. A. Zalgaller, Convex polyhedra with regular faces, Seminar in Mathematics Steklov Math. Institute, Leningrad, vol 2; Consultants Bureau, New York, 1969. [Zie95] G. M. Ziegler, Lectures on polytopes, Graduate texts in mathematics 152, Springer-Verlag 1995. Authors’ addresses: Michel Marie Deza LIENS, DMI Ecole Normale Supérieure 45 rue d’Ulm 75230 Paris Cedex 05 France e-mail: [email protected] Dmitrii V. Pasechnik Dept. of Computer Science Utrecht University PO Box 80.089 3508 TB Utrecht The Netherlands e-mail: [email protected] 13