On the number of distinct minimal clique partitions and clique covers of a line graph

Abstract

Let G be a line graph. Orlin determined the clique covering and clique partition numbers cc( G) and cp( G). We obtain a constructive proof of Orlin's result and in doing so we are able to completely enumerate the number of distinct minimal clique covers and partitions of G, in terms of easily calculable parameters of G. We apply our results to give a new proof of Whitney's Theorem: if G and H are graphs, neither of which is K 3, then G and H are isomorphic if and only if their line graphs are isomorphic.