On the intersection number of a graph

Abstract

We find an expression of the intersection number of a graph in terms of the minimum number of complete subgraphs that form a covering of the graph. This provides us with a uniform approach to studying properties of the intersection number of a graph. We distinguish the class of graphs for which the intersection number is equal to the least number of cliques covering the graph. It is proved that the intersection number of a complete r-partite graph r x K 2 is equal to the least n such that r n 1 OEn=2 1. It is proved that the intersection number of the graph r x K 2 C K m is equal to the least n such that m C r 2 n 1 , r n 1 OEn=2 1 . Formulas for the intersection numbers of the graphs rC 4 , r Chain.3/, r.C 4 C K m /, rW 5 are obtained.