On Equitable Colorings of Sparse Graphs

Abstract

A graph is equitably k-colorable if G has a proper vertex k-coloring such that the sizes of any two color classes differ by at most one. Chen, Lih and Wu conjectured that any connected graph G with maximum degree $$\Delta $$ distinct from the odd cycle, the complete graph $$K_{\Delta +1}$$ and the complete bipartite graph $$K_{\Delta ,\Delta }$$ are equitably m-colorable for every $$m\ge \Delta $$ . Let $${\mathcal {G}}_k$$ be the class of graphs G such that $$e(G')\le k (v(G')-2)$$ for every subgraph $$G'$$ of G with order at least 3. In this paper, it is proved that any graph in $${\mathcal {G}}_4$$ with maximum degree $$\Delta \ge 17$$ is equitably m-colorable for every $$m\ge \Delta $$ . As corollaries, we confirm Chen–Lih–Wu Conjecture for 1-planar graphs, 3-degenerate graphs and graphs with maximum average degree less than 6, provided that $$\Delta \ge 17$$ .