For a graph $G$, the \emph{equitable chromatic number} of $G$, denoted by $χ_e(G)$, is the smallest integer $k$ such that $G$ admits a proper $k$-coloring whose color classes differ in size by at most one. We prove that for every $ζ>41/2$, there exists a constant $c=c(ζ)\in\mathbb{N}$ such that every bipartite graph $G$ with maximum degree $Δ(G)\ge c$ and $|V(G)|\ge ζΔ(G)$ satisfies $χ_e(G)\le \left\lceilΔ(G)/2\right\rceil+1$. The leading term $Δ(G)/2$ in this bound is best possible for upper bounds stated solely in terms of $Δ(G)$ for bipartite graphs. Our proof yields an $O(|V(G)|^2)$-time algorithm for constructing such a coloring.

翻译：对于图 $G$，其\emph{均衡色数}（记作 $χ_e(G)$）是满足以下条件的最小整数 $k$：$G$ 存在一个正常 $k$-着色，且其颜色类的大小至多相差1。我们证明，对于每个 $ζ>41/2$，存在常数 $c=c(ζ)\in\mathbb{N}$，使得每个最大度 $Δ(G)\ge c$ 且 $|V(G)|\ge ζΔ(G)$ 的二部图 $G$ 满足 $χ_e(G)\le \left\lceilΔ(G)/2\right\rceil+1$。该界中的首项 $Δ(G)/2$ 对于仅以 $Δ(G)$ 表述的二部图的上界而言是最优的。我们的证明给出了一个构造此类着色的 $O(|V(G)|^2)$ 时间算法。