Disjoint cliques and disjoint maximal independent sets of vertices in graphs

Disjoint cliques and disjoint maximal independent sets of vertices in graphs

In this paper, we find lower bounds for the maximum and minimum numbers of cliques in maximal sets of pairwise disjoint cliques in a graph . By complementation, these yield lower bounds for the maximum and minimum numbers of independent sets in maximal sets of pairwise disjoint maximal independent sets of vertices in a graph . In the latter context, we show by examples that one of our bounds is...

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 ...

The Disjoint Cliques Problem

On the maximal number of vertices covered by disjoint cycles

Let k, t and n be three integers with t ~ 2, k ~ 2t and n ~ 3t. We conjecture that if G is a graph of order n with minimum degree at least k, then G contains t vertex-disjoint cycles covering at least min(2k, n) vertices of G. We will show the conjecture to be true for t = 2.

On the problems of edge disjoint cliques in graphs