发表机构
Middle Tennessee State University; Augustana College（中田纳西州立大学; 奥古纳学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对图的D-色指数的Brooks型上界问题，采用概率方法，首次改进了此前由贪心着色得到的上界，证明了最大度足够大时D-色指数不超过带正系数修正的二次上界。
AI 中文摘要
对于图$G$，若$G$的一个正常边着色使得$G$的每个钻石子图都是彩虹的，则称其为D-着色。设$χ'_D(G)$为$G$的D-色指数，即满足$G$可使用$k$种颜色进行D-着色的最小整数$k$。令$Δ$为$G$的最大度。目前已知的$χ'_D(G)$唯一的Brooks型上界为$\frac{9}{16}Δ^2 + \frac{1}{2}Δ$，由贪心着色得到。本文利用概率方法，证明了存在$c > 0$，使得当$Δ$足够大时，$χ'_D(G) \le (1-c)\frac{9}{16}Δ^2$，首次改进了该上界。
英文摘要
For a graph $G$, a proper edge coloring of $G$ is called a D-coloring if every diamond subgraph of $G$ is rainbow. Let $χ'_D(G)$ be the D-chromatic index of $G$, which is the smallest integer $k$ such that $G$ admits a D-coloring with $k$ colors. Let $Δ$ be the maximum degree of $G$. The only known Brooks-type upper bound on $χ'_D(G)$ is $\frac{9}{16}Δ^2 + \frac{1}{2}Δ$, given by a greedy coloring. In this paper, using a probabilistic method, we obtain the first improvement upon this upper bound by proving that $χ'_D(G) \le (1-c)\frac{9}{16}Δ^2$ for some $c > 0$ and sufficiently large $Δ$.
