Abstract
Corrádi and Hajnal proved that for every k ≥ 1 and n ≥ 3k, every n-vertex graph with minimum degree at least 2k contains k vertex-disjoint cycles. This implies that every 3k-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 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.
Original language | English (US) |
---|---|
Pages (from-to) | 497-512 |
Number of pages | 16 |
Journal | Combinatorica |
Volume | 35 |
Issue number | 4 |
DOIs | |
State | Published - Aug 22 2015 |
Keywords
- 05C15
- 05C35
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics
- Computational Mathematics