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DM
1998

An Euler-type formula for median graphs

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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...
Sandi Klavzar, Henry Martyn Mulder, Riste Skrekovs
Added 22 Dec 2010
Updated 22 Dec 2010
Type Journal
Year 1998
Where DM
Authors Sandi Klavzar, Henry Martyn Mulder, Riste Skrekovski
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