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arXiv · 2608.14517

A disproof of a gap-one conjecture for the equitable chromatic number of block graphs

Juho Lauri
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Abstract

For a graph $G$, let $L(G)=\max\{\omega(G),\lceil (|V(G)|+1)/(\alpha_{\min}(G)+1)\rceil\}$, where $\omega(G)$ is the clique number and $\alpha_{\min}(G)$ is the minimum, over all vertices $v$, of the largest size of an independent set containing $v$. Dybizba\'nski, Furma\'nczyk, and Mkrtchyan (Discrete Appl. Math. 354 (2024), 15--28) conjectured that every block graph $G$ satisfies $L(G)\leq\chi_{=}(G)\leq L(G)+1$, where $\chi_{=}(G)$ is the equitable chromatic number of $G$. We disprove this conjecture in a strong form. For every pair of integers $d\geq 2$ and $k\geq 4d-1$, we construct a connected block graph $G_{d,k}$ such that $L(G_{d,k})=k$ and $\chi_{=}(G_{d,k})=k+d$. Thus the difference $\chi_{=}(G)-L(G)$ is unbounded on connected block graphs.

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Juho Lauri. 2026-08-14. A disproof of a gap-one conjecture for the equitable chromatic number of block graphs. https://arxiv.org/abs/2608.14517

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