Extremal clique coverings of complementary graphs

Let G be a graph on n vertices and let G be its complement in K,,, the complete graph on n vertices . If f is a real valued function defined on graphs, what are the extreme values of f (G) +f (G) and f (G) f (G)? E. A. Nordhaus and J . W. Gaddum (see e .g . [5]) considered those questions when the function is the chromatic number . D . Taylor, R. D . Dutton and R. C. Brigham [5] studied the questions for several other functions. One of those is the clique covering number . That is cc(G), the least number of complete subgraphs (cliques) of G necessary to cover the edge set of G . We continue their investigation . We also consider the questions for another function the clique partition number. That is cp(G), the least number of cliques needed to partition the edge set of G. In Theorem 1, we establish the right inequality of [n 2/4]+2_-max {cc(G)+ +cc(G))~(n2/4)(1 +0(1)) where the maximum is taken over all €raphs G on n vertices . The bipartite graph Kt„t2t .fn/21 assumes the lower bound . In Theorem 2 we modify the proof of Theorem 1 to show that max {cc(G)cc(G))-(n4/256)(1+0(l)), where the maximum is taken over all nvertex graphs G . D. Taylor et al . [5, Theorem 51 gave an example of a graph F for which cc(F)cc(F)=n 2(n+8)2/256 . The graph F is obtained from two copies Al and A2 of K„, 4 and two copies A2 and A3 of KnI , by joining each vertex of A ; to each vertex of Ai+i (i=1, 2 and 3). When n is not divisible by 4 the construction can be modified to yield a similar graph. Hence Theorem 2 establishes the conjecture made in [51, that max {cc(G)cc(G))-n4/256 where the maximum is taken over all n-vertex graphs G.