THE EQUIVALENCE NUMBER OF A LINE GRAPH

The chromatic index of a graph G is most often defined to be the minimum size of a partition of the edge set of G into matchings. An equivalent but different de finition is the minimum size of a cover of the edge set of G by matchings. We consider the analogous problem of covering the edge set of G by subgraphs that are vertex-disjoint unions of cliques, known as equivalence graphs. The minimum size of such a cover is the equivalence number of G. We compute the equivalence number of the line graph of a clique on at most 12 vertices. We also construct a particular type of cover to show that, for all graphs G on at most n vertices, the equivalence number of the line graph of G has an upper bound on the order of log n. Finally, we show that if G is a clique on 13 vertices then the minimum size of this particular cover is 5.