An Euler-type formula for median graphs

Let G be a median graph on n vertices and m edges and let k be the number of equivalence classes of the Djoković’s relation Θ defined on the edge-set of G. Then 2n − m − k ≤ 2. Moreover, 2n − m − k = 2 if and only if G is cube-free. A median graph is a connected graph such that, for every triple of vertices u, v, w, there is a unique vertex x lying on a geodesic (i.e. shortest path) between each pair of u, v, w. By now, the class of median graphs is well studied and a rich structure theory is available, see e.g. [5]. In this note, we present an Euler-type formula for median graphs, which involves the number of vertices n, the number of edges m, and the number of Θ-classes k (or, equivalenty, the number of cutsets in the cutset coloring, cf. [6,7]). The formula is an inequality, where equality is attained if and only if the median graph is cube-free. 1 Supported by the Ministry of Science and Technology of Slovenia under the grant J1-7036 and by the SWON, the Netherlands 2 Supported by the Ministry of Science and Technology of Slovenia and by the NUFFIC, the Netherlands. 3 Supported by the Ministry of Science and Technology of Slovenia under the grant J1-7036. Preprint submitted to Elsevier 1 October 2010 For u, v ∈ V (G) let dG(u, v) denote the length of a shortest path in G from u to v. A subgraph H of G is convex, if for any u, v ∈ V (H), all shortest paths between u and v belong to H . Clearly, a convex subgraph is connected. The Djoković’s relation Θ introduced in [1] is defined on the edge-set of a graph in the following way. Two edges e = xy and f = uv of a graph G are in relation Θ if dG(x, u) + dG(y, v) 6= dG(x, v) + dG(y, u) . Clearly, Θ is reflexive and symmetric. If G is bipartite, relation Θ can be rewritten as follows: e = xy and f = uv are in relation Θ if d(x, u) = d(y, v) and d(x, v) = d(y, u) . Among bipartite graphs, Θ is transitive precisely for partial cubes (i.e. isometric subgraphs of hypercubes), as proved by Winkler in [9]. Since median graphs form a subclass of partial cubes, Θ is in particular transitive for median graphs. Thus Θ is a congruence on median graphs. For more information on Θ we refer to [2,3]. Let G = (V, E) be a connected graph. For two subgraphs G1 = (V1, E1) and G2 = (V2, E2) of G, the intersection G1∩G2 is the subgraph of G with vertexset V1 ∩ V2 and edge-set E1 ∩ E2, and the union G1 ∪ G2 is the subgraph of G with vertex-set V1 ∪ V2 and edge-set E1 ∪ E2. A convex cover G1, G2 of G consists of two convex subgraphs G1 and G2 of G such that G0 = G1 ∩ G2 is non-empty and G = G1 ∪ G2. Note that there are no edges between G1 − G2 and G2 − G1, and that G0 is convex. Let G be a connected graph, and let G′1, G ′ 2 be a convex cover of G ′ with G′0 = G ′ 1 ∩ G ′ 2. The expansion of G ′ with respect to G′1, G ′ 2 is the graph G constructed as follows. Let Gi be an isomorphic copy of G ′ i, for i = 1, 2, and, for any vertex u in G′0, let ui be the corresponding vertex in Gi, for i = 1, 2. Then G is obtained from the disjoint union G1 ∪ G2, where u1 and u2 are joined by an edge for each u in G′0. For example, if G ′ = G′1 = G ′ 2 = Qn, then G = Qn+1, and if G ′ is a tree and G′1 = G ′ and G′2 = {u }, then G is obtained from G by adding a new vertex pending at u. We call G1, G2 a split in G. Note that the above notion of expansion is called convex Cartesian expansion in the much more general setting of [8]. We will say that G is obtained from a graph H by an expansion procedure if we obtain G from H by a sequence of expansions. An important tool in the study of median graphs is the following theorem from [6,7]. Theorem 1 A graph is a median graph if and only if it can be obtained from the one vertex graph by an expansion procedure.