稠密与完全多部图中的锐利彩虹路径覆盖

中文摘要 AI 辅助

在正常边染色的图中，如果一条路径的边两两颜色不同，则称其为彩虹路径。对于图$G$的正常边染色$c$，令$\operatorname{rpc}(G,c)$为覆盖$E(G)$所需的最少彩虹路径数，并令$\operatorname{rpc}(G)$为$G$的所有正常边染色中$\operatorname{rpc}(G,c)$的最大值。我们证明，对于每个固定的$0<\alpha<1$，每个最小度至少为$\alpha n$的$n$顶点正常边染色图满足$\operatorname{rpc}(G,c)\leq(1+o(1))n/2$，其中系数$1/2$是最优的。我们还渐近确定了每个完全多部图的$\operatorname{rpc}(G)$。若$G=K_{n_1,\ldots,n_r}$的阶为$n$，最大和最小部分大小分别为$M$和$s$，则对所有部分数量和大小选择一致地有$\operatorname{rpc}(G)=(1+o(1))\max\{\min\{\lfloor n/2\rfloor,n-M\},(n-s)/2\}$。证明结合了全局彩虹线性森林的伪随机打包与稠密部分的分解及对任意稠密图的指定回避，以及对完全多部图使用预留连接器和直接主部论证。

英文摘要

A path in a properly edge-colored graph is rainbow if its edges have pairwise distinct colors. For a proper edge-coloring $c$ of a graph $G$, let $\operatorname{rpc}(G,c)$ be the minimum number of rainbow paths needed to cover $E(G)$, and let $\operatorname{rpc}(G)$ be the maximum of $\operatorname{rpc}(G,c)$ over all proper edge-colorings of $G$. We prove that, for every fixed $0<α<1$, every properly edge-colored $n$-vertex graph with minimum degree at least $αn$ satisfies $\operatorname{rpc}(G,c)\leq(1+o(1))n/2$, where the coefficient $1/2$ is best possible. We also determine $\operatorname{rpc}(G)$ asymptotically for every complete multipartite graph. If $G=K_{n_1,\ldots,n_r}$ has order $n$ and largest and smallest part sizes $M$ and $s$, respectively, then, uniformly over all choices of the number and sizes of the parts, $\operatorname{rpc}(G)=(1+o(1))\max\{\min\{\lfloor n/2\rfloor,n-M\},(n-s)/2\}$. The proof combines pseudorandom packings of globally rainbow linear forests with a decomposition into dense parts and prescribed avoidance for arbitrary dense graphs, and with reserved connectors and a direct dominant-part argument for complete multipartite graphs.

发表机构

• Universidade Federal de Pernambuco（伯南布哥联邦大学）
• Guizhou Minzu University（贵州民族大学）
• Universidade Federal de Alagoas（阿拉戈阿斯联邦大学）

机构由 AI 辅助整理，请以论文原文为准。

补充信息

备注

25 pages