A refinement of a result of Corrádi and Hajnal

Corrádi and Hajnal proved that for every k ≥ 1 and n ≥ 3k, every graph with minimum degree at least 2k contains k vertex-disjoint cycles. This implies that every 3kvertex graph with maximum degree at most k − 1 has an equitable k-coloring. We prove that for s ∈ {3, 4} if an sk-vertex graph G with maximum degree at most k has no equitable k-coloring, then G either contains Kk+1 or k is odd and G contains Kk,k. This refines the above corollary of the Corrádi-Hajnal Theorem and also is a step toward the conjecture by Chen, Lih, and Wu that for r ≥ 3, the only connected graphs with maximum degree at most r that are not equitably r-colorable are Kr,r (for odd r) and Kr+1. Mathematics Subject Classification: 05C15, 05C35.