## Abstract

Corrádi and Hajnal proved that for every *k* ≥ 1 and *n* ≥ 3*k*, every *n*-vertex graph with minimum degree at least 2*k* contains *k* vertex-disjoint cycles. This implies that every 3*k*-vertex 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 *K*
_{
k+1} or *k* is odd and *G* contains *K*
_{
k,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 *K*
_{
r,r
} (for odd *r*) and *K*
_{
r+1}.

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## Additional information

Research of this author is supported in part by NSA grant H98230-12-1-0212.

Research of this author is supported in part by NSF grant DMS-0965587 and by grants 12-01-00448 and 12-01-00631 of the Russian Foundation for Basic Research.

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Kierstead, H.A., Kostochka, A.V. A refinement of a result of Corrádi and Hajnal.
*Combinatorica* **35, **497–512 (2015). https://doi.org/10.1007/s00493-014-3059-6

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### Mathematics Subject Classification (2000)

- 05C15
- 05C35