Equitable coloring of sparse graphs

Abstract

An equitable coloring of a graph is a proper coloring where the sizes of any two distinct color classes differ by at most one. The celebrated Chen-Lih-Wu Conjecture (CLWC for short) states that every connected graph $G$ that is neither an odd cycle, a $K_r$, nor a $K_{2m+1,2m+1}$ has an equitable $\Delta(G)$-coloring. A graph $G$ is in $\mathcal{G}_{m_1,m_2}$ if for all $H\subseteq G$, $\lVert H \rVert\leq m_1|H|$, and if $H$ is bipartite, then $\lVert H \rVert\leq m_2|H|$. In this paper, we confirm CLWC for all graphs $G$ in $\mathcal{G}_{m_1, m_2}$ provided that $m_1\leq 1.8m_2$ and $\Delta(G)\geq \frac{2m_1}{1-\beta}$, where $\beta$ is a real root of $2m_2(1-x)(1+x)^2-m_1x(2+x)$. By specializing to the case $m_1 = m_2 = d$, we deduce that every $d$-degenerate graph $G$ with $\Delta(G) \geq 6.21d$ admits an equitable $r$-coloring for all $r \geq \Delta(G)$, thereby improving the previous best-known lower bound of $10d$ on $\Delta(G)$ established by Kostochka and Nakprasit in 2005. A graph is $k$-planar if it can be drawn in the plane so that each edge is crossed at most $k$ times. CLWC had been confirmed for planar graphs $G$ with $\Delta(G) \geq 8$ (Kostochka, Lin, and Xiang, 2024) and for $1$-planar graphs $G$ with $\Delta(G) \geq 13$ (Cranston and Mahmoud, 2025). As an immediate application of our main result, we extend this confirmation to all $k$-planar graphs $G$ with $k \geq 2$ and $\Delta(G) \geq \sqrt{383k}$.