Corrádi and Hajnal (Acta Math Acad Sci Hung 14:423–439, 1963) proved that for all $$k\ge 1$$k≥1 and $$n\ge 3k$$n≥3k, every (simple) graph G on n vertices with minimum degree $$\delta (G)\ge 2k$$δ(G)≥2k 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 $$k\ge 1$$k≥1 and $$n\ge 3k$$n≥3k, every graph G on n vertices contains k disjoint cycles, provided that $$d(x)+d(y)\ge 4k-1$$d(x)+d(y)≥4k-1 for all distinct nonadjacent vertices x, y. Very recently, it was refined for $$k\ge 3$$k≥3 and $$n\ge 3k+1$$n≥3k+1: If G is a graph on n vertices such that $$d(x)+d(y)\ge 4k-3$$d(x)+d(y)≥4k-3 for all distinct nonadjacent vertices x, y, then G has k vertex-disjoint cycles if and only if the independence number $$\alpha (G)\le n-2k$$α(G)≤n-2k and G is not one of two small exceptions in the case $$k=3$$k=3. But the most difficult case, $$n=3k$$n=3k, 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.

中文翻译：

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锐化 Corrádi-Hajnal 定理的矿石类型版本

Corrádi 和 Hajnal (Acta Math Acad Sci Hung 14:423–439, 1963) 证明对于所有 $$k\ge 1$$k≥1 和 $$n\ge 3k$$n≥3k，每个（简单的）图最小度数 $$\delta (G)\ge 2k$$δ(G)≥2k 的 n 个顶点上的 G 包含 k 个不相交的循环。度数界限是尖锐的。Enomoto 和 Wang 证明了 Corrádi–Hajnal 定理的以下 Ore-type 改进：对于所有 $$k\ge 1$$k≥1 和 $$n\ge 3k$$n≥3k，n 个顶点上的每个图 G 包含k 个不相交的循环，假设对于所有不同的非相邻顶点 x, y，$$d(x)+d(y)\ge 4k-1$$d(x)+d(y)≥4k-1。最近，它被改进为 $$k\ge 3$$k≥3 和 $$n\ge 3k+1$$n≥3k+1：如果 G 是 n 个顶点上的图，使得 $$d(x )+d(y)\ge 4k-3$$d(x)+d(y)≥4k-3 对于所有不同的非相邻顶点 x, y, 则 G 有 k 个顶点不相交环当且仅当独立数 $$\alpha (G)\le n-2k$$α(G)≤n-2k 并且 G 不是情况 $ 中的两个小例外之一$k=3$$k=3。但是最困难的情况，$$n=3k$$n=3k，没有处理。在这种情况下，异常图更多，语句更复杂，一些证明不起作用。在本文中，我们解决了这个困难的情况，并获得了 Corrádi-Hajnal 定理的矿石型版本的极值图的全貌。由于 3k 顶点图 G 中的任何 k 个不相交循环都必须是 3 个循环，因此这些 k 个循环的存在等价于 G 的补集的公平 k 着色的存在。我们的证明使用了公平着色的语言，