A note on the orientation covering number
Abstract
Given a graph G, its orientation covering number σ(G) is the smallest non-negative integer k with the property that we can choose k orientations of G such that whenever x, y, z are vertices of G with xy,xz∈ E(G) then there is a chosen orientation in which both xy and xz are oriented away from x. Esperet, Gimbel and King showed that σ(G)≤ σ(K(G)), where (G) is the chromatic number of G, and asked whether we always have equality. In this note we prove that it is indeed always the case that σ(G)=σ(K(G)). We also determine the exact value of σ(Kn) explicitly for `most' values of n.
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