Sharpening an Ore-type version of the Corrádi–Hajnal theorem

Corrádi and Hajnal (Acta Math Acad Sci Hung 14:423–439, 1963) proved that for all k≥ 1 and n≥ 3 k, every (simple) graph G on n vertices with minimum degree δ(G) ≥ 2 k contains k disjoint cycles. The degree bound is sharp. Enomoto and Wang proved the following Ore-type refinement of the Corrádi–Hajnal theorem: For all k≥ 1 and n≥ 3 k, every graph G on n vertices contains k disjoint cycles, provided that d(x) + d(y) ≥ 4 k- 1 for all distinct nonadjacent vertices x, y. Very recently, it was refined for k≥ 3 and n≥ 3 k+ 1 : If G is a graph on n vertices such that d(x) + d(y) ≥ 4 k- 3 for all distinct nonadjacent vertices x, y, then G has k vertex-disjoint cycles if and only if the independence number α(G) ≤ n- 2 k and G is not one of two small exceptions in the case k= 3. But the most difficult case, n= 3 k, was not handled. In this case, there are more exceptional graphs, the statement is more sophisticated, and some of the proofs do not work. In this paper we resolve this difficult case and obtain the full picture of extremal graphs for the Ore-type version of the Corrádi–Hajnal theorem. Since any k disjoint cycles in a 3k-vertex graph G must be 3-cycles, the existence of such k cycles is equivalent to the existence of an equitable k-coloring of the complement of G. Our proof uses the language of equitable colorings, and our result can be also considered as an Ore-type version of a partial case of the Chen–Lih–Wu Conjecture on equitable colorings.