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arXiv · 2609.07014

Bounds for the Vertex Chromatic Number of Connected Triangle-Free Graphs

Saieed Akbari·Arash Beikmohammadi
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Abstract

It was recently shown that every connected graph of order $n \geq 5$ and size $m$ satisfies $\chi(G) \leq \left\lceil \frac{m}{\sqrt{n}} \right\rceil$, and it was asked whether the stronger inequality $\chi(G) \leq \left\lceil \frac{m}{2\sqrt{n}} \right\rceil + 1$ holds for every connected triangle-free graph. In this paper, we answer this question in the affirmative. In fact, we prove that every connected triangle-free graph $G$ with $G \not\cong C_5$ satisfies $\chi(G) \leq \left\lceil \frac{m}{\sqrt{5.5n}} \right\rceil + 1$, where the constant $\sqrt{5.5}$ cannot be replaced by any constant greater than or equal to $\sqrt{6}$, and the equality holds for the Gr\"{o}tzsch graph and for every odd cycle of length between $7$ and $21$.

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Saieed Akbari, Arash Beikmohammadi. 2026-09-07. Bounds for the Vertex Chromatic Number of Connected Triangle-Free Graphs. https://arxiv.org/abs/2609.07014

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