Equitable coloring of large bipartite graphs

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.