Disjoint cliques in regular graphs of degree seven and eight

This paper is motivated by Berge’s conjecture that every undirectefl regular graph, except the complement of a complete graph, has at least two disjoint maximal independent sets of vertices. By complementation this is equivalent to the conjecture that every incomplete regular graph has two or more disjoint cliques. In [l], we established this form of the conjecture for regular graphs of degree < 6. Here we prove that incomplete graphs of degree 7 or 8 possess disjoint cliques. Any terminology or notation undefined in this paper may be found in Harary’s book [2].