Equitable coloring of large bipartite graphs

TL;DR

This paper establishes an upper bound on the equitable chromatic number of large bipartite graphs with high maximum degree, and provides an efficient algorithm for constructing such colorings.

Contribution

It proves a new bound on the equitable chromatic number for bipartite graphs with large maximum degree and offers a polynomial-time coloring algorithm.

Findings

Bound $oxed{ ext{ceil}(rac{ riangle}{2})+1}$ for equitable chromatic number in large bipartite graphs.

The bound is tight for graphs depending only on maximum degree.

Provides an $O(|V(G)|^2)$-time algorithm for equitable coloring.

Abstract

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.