发表机构
Hong Kong University of Science and Technology（香港科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究无（\(P_2\cup P_3\)）图的色数界，通过证明\(\chi(G)\leq \binom{k + 2}{3} - \binom{k - 1}{2}\)给出改进方法，相比之前结果节省颜色并消除二次项，是对该图类界的首次改进。
AI 中文摘要
设\(G\)是无（\(P_2\cup P_3\)）图，且\(k = \omega(G) \geq 4\)。我们证明了\(\chi(G)\leq \binom{k + 2}{3} - \binom{k - 1}{2} = \frac{k^3 + 11k - 6}{6}\)。此前Bharathi和Choudum给出的该无限制图类的最佳已知一般界是\(\chi(G)\leq\binom{k + 2}{3}\)。据我们所知，这是对其界的首次改进，适用于所有无（\(P_2\cup P_3\)）图。结果节省了\(\binom{k - 1}{2}\)种颜色，消除了二次项。
英文摘要
We prove that every \((P_2\cup P_3)\)-free graph \(G\), with \(k=ω(G)\), satisfies \[ χ(G)=O\!\left(k^3\frac{\ln\ln k}{\ln k}\right). \] Thus the class admits an \(o(k^3)\) binding function. We also prove the explicit bound \[ χ(G)\le \frac{k^3}{8}+\frac{5k^2}{4}-4k+8 \qquad(k\ge8), \] with a sharper formula for odd \(k\). A further refinement gives an explicit cubic bound with leading coefficient \(104/837<1/8\).
Comments22 pages. Substantially rewritten and retitled, with a new $O(k^3\ln\ln k/\ln k)$ chromatic bound for the entire $(P_2\cup P_3)$-free class ($k=ω(G)$) and sharper explicit cubic bounds. The earlier row estimates and the quadratic bound under a rank-gap condition are not reproduced here; see v3 for those results
