Covering line graphs with equivalence relations

An equivalence graph is a disjoint union of cliques, and the equivalence number eq(G) of a graph G is the minimum number of equivalence subgraphs needed to cover the edges of G. We consider the equivalence number of a line graph, giving improved upper and lower bounds. 1/3 log(2) log(2) chi (G) < eq(L(G)) <= 2 log(2) log(2) chi (G) + 2. This disproves a recent conjecture that eq(L(G)) is at most three for triangle-free G: indeed it can be arbitrarily large To bound eq(L(G)) we bound the closely related invariant sigma(G), which is the minimum number of orientations of G such that for any two edges e, f incident to some vertex v, both e and f are oriented out of v in some orientation When G is triangle-free, sigma(G) = eq(L(G)). We prove that even when G is triangle-free, It is NP-complete to decide whether or not sigma(G) <= 3. (C) 2010 Elsevier B.V All rights reserved.