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Completely independent critical cliques

JOHN J. LATTANZIO1
1Department of Mathematics Indiana University of Pennsylvania, Indiana, PA 15701

Abstract

If K is an r-clique of G and χ(G) decreases by r upon the removal of all of the vertices in K, then K is called a critical r-clique. Two critical cliques are completely independent provided that no vertex in one clique is adjacent to a vertex from the other. An infinite family of graphs is constructed which demonstrates that for every s,tN, there exists a vertex critical graph which admits a critical s-clique and a critical t-clique that are completely independent.