Some bounds on convex combinations of ω and χ for decompositions into many parts

Some bounds on convex combinations of ω and χ for decompositions into many parts Abstract A k–decomposition of the complete graph K n is a decomposition of K n into k spanning subgraphs G 1 ,. .. , G k. For a graph parameter p, let p(k; K n) denote the maximum of k j=1 p(G j) over all k–decompositions of K n. It is known that χ(k; K n) = ω(k; K n) for k ≤ 3 and conjectured that this equality holds for all k. In an attempt to get a handle on this, we study convex combinations of ω and χ; namely, the graph parameters A r (G) = (1 − r)ω(G) + rχ(G) for 0 ≤ r ≤ 1. It is proven that A r (k; K n) ≤ n + k 2 for small r. In addition, we prove some generalizations of a theorem of Kostochka, et al. [1].