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 Djokovi´c’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 Netherlan...

• Type: Journal
• Year: 1998
• Where: DM
• Authors: Sandi Klavzar, Henry Martyn Mulder, Riste Skrekovski
• Keywords: Djokovi´c’s Relation, DM 1998, Equivalence Classes, Median Graph