Equitable coloring of large bipartite graphs
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917387637358592 |
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| author | Nikabadi, Amir |
| author_facet | Nikabadi, Amir |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_05146 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Equitable coloring of large bipartite graphs Nikabadi, Amir Combinatorics Discrete Mathematics 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. |
| title | Equitable coloring of large bipartite graphs |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2604.05146 |