AI 中文总结
本文证明了连通无三角形图（除五圈外）的色数上界为 $\lceil m/\sqrt{5.5n}\rceil+1$，并指出常数最优，等号在 Grötzsch 图和特定奇圈上成立。
AI 中文摘要
最近有研究证明，每个阶数 $n \geq 5$、边数 $m$ 的连通图都满足 $\chi(G) \leq \left\lceil \frac{m}{\sqrt{n}} \right\rceil$，并提出了一个更强的不等式是否成立的问题：对于每个连通的无三角形图，是否有 $\chi(G) \leq \left\lceil \frac{m}{2\sqrt{n}} \right\rceil + 1$？本文对此问题给出了肯定回答。实际上，我们证明了每个满足 $G \not\cong C_5$ 的连通无三角形图 $G$ 都满足 $\chi(G) \leq \left\lceil \frac{m}{\sqrt{5.5n}} \right\rceil + 1$，其中常数 $\sqrt{5.5}$ 不能被任何大于或等于 $\sqrt{6}$ 的常数所替代，且等号在 Grötzsch 图和所有长度在 $7$ 到 $21$ 之间的奇圈上成立。
英文摘要
It was recently shown that every connected graph of order $n \geq 5$ and size $m$ satisfies $χ(G) \leq \left\lceil \frac{m}{\sqrt{n}} \right\rceil$, and it was asked whether the stronger inequality $χ(G) \leq \left\lceil \frac{m}{2\sqrt{n}} \right\rceil + 1$ 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 $χ(G) \leq \left\lceil \frac{m}{\sqrt{5.5n}} \right\rceil + 1$, 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 Grötzsch graph and for every odd cycle of length between $7$ and $21$.
