发表机构
University of Engineering and Technology (UTEC)（工程与技术大学（UTEC））
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究正则图的动态着色，证明无爪三次图的三动态色数不超过6且界紧，改进哈密顿情形至5，并精确计算循环图C_p(1,3)的r动态色数。
AI 中文摘要
图$G$的$r$-动态着色是一种正常顶点着色，其中每个顶点$v$在其邻域中至少看到$\min\{r,d(v)\}$种不同颜色。此类着色所需的最少颜色数称为$r$-动态色数$\chi_r(G)$。我们研究正则图的动态着色。一个直接观察表明，对于任何$r$-正则图$G$，有$\chi_r(G)=\chi(G^2)$。我们证明，对于每个无爪三次图$G$，有$\chi_3(G)=\chi(G^2)\le6$，且该界是紧的。对于哈密顿无爪三次图，除四个明确例外外，我们将界改进为$5$。我们还精确确定了$4$-正则循环图$C_p(1,3)$对每个$r\in\{2,3,4\}$的$r$-动态色数。特别地，$r=4$的情形确定了$\chi(C^2_p(1,3))$。
英文摘要
An $r$-dynamic coloring of a graph $G$ is a proper vertex coloring in which every vertex $v$ sees at least $\min\{r,d(v)\}$ distinct colors in its neighborhood. The minimum number of colors in such a coloring is the $r$-dynamic chromatic number $χ_r(G)$. We study dynamic colorings of regular graphs. A straighforward observation shows that $χ_r(G)=χ(G^2)$, for any $r$-regular graph $G$. We prove that $χ_3(G)=χ(G^2)\le6$ for every claw-free cubic graph $G$, and the bound is sharp. For Hamiltonian claw-free cubic graphs, we improve the bound to $5$ apart from four explicit exceptions. We also determine exactly the $r$-dynamic chromatic number of the $4$-regular circulant graph $C_p(1,3)$ for each $r\in\{2,3,4\}$. In particular, the case $r=4$ determines $χ(C^2_p(1,3))$.
Comments15 pages
