Equitable coloring of large bipartite graphs

For a graph $G$, the \emph{equitable chromatic number} of $G$, denoted by $\chi_e(G)$, 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 $\zeta>41/2$, there exists a constant $c=c(\zeta)\in\mathbb{N}$ such that every bipartite graph $G$ with maximum degree $\Delta(G)\ge c$ and $|V(G)|\ge \zeta\Delta(G)$ satisfies $\chi_e(G)\le \left\lceil\Delta(G)/2\right\rceil+1$.

The leading term $\Delta(G)/2$ in this bound is best possible for upper bounds stated solely in terms of $\Delta(G)$ for bipartite graphs.

Our proof yields an $O(|V(G)|^2)$-time algorithm for constructing such a coloring.