A Refinement of the Triangle Version of the Corradi-Hajnal Theorem
Kostochka, Alexandr V.
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An important part of the Corradi-Hajnal Theorem says that if n = 3k, then every n-vertex graph G with minimum degree at least 2k=2n/3 contains k vertex-disjoint triangles. The restriction on the minimum degree is sharp. This is equivalent (by switching to the complement) to the statement that every n-vertex graph H with maximum degree at most k-1 has an equitable k-coloring, that is a proper coloring of vertices of H with k colors such that the sizes of color classes differ by at most 1. In 1970, Hajnal and Szemeredi generailzed this result by proving the conjecture of Erdos that every graph with maximum degree at most r has an equitable r+1-coloring. In this talk, we prove a Brooks-type result describing for r\geq n/4 all n-vertex graphs with maximum degree at most r that do not admit an equitable r-coloring. In particular, we conclude that for k\geq 6, at most one 3k-vertex graph with minimum degree at least 2k-1 and independence number at most k does not contain k vertex-disjoint triangles. This is joint work with H. A. Kierstead.