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 log2 log2 χ(G) < eq(L(G)) ≤ 2 log2 log2 χ(G) + 2. This disproves a recent conjecture that eq(L(G)) is at most three for trianglefree G; indeed it can be arbitrarily large. To bound eq(L(G)) we bound the closely-related invariant σ(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, σ(G) = eq(L(G)). We prove that even when G is triangle-free, it is NP-complete to decide whether or not σ(G) ≤ 3.