AI 中文总结
本文研究书图B_n的弱化Gallai-Ramsey数，扩展了前人t=3时的结果，确定了t>3时gr^3_2(B_n)在5≤n≤15的值，并给出该类数的一般下界。
AI 中文摘要
对于1≤s<t及任意图G，弱化Gallai-Ramsey数gr^t_s(G)定义为最小正整数p，使得完全图K_p的任意t边着色（即不含彩虹三角形的t着色）都包含一个与G同构的子图，其边最多使用s种颜色。对于书图B_n:=K_2+nK_1，Jakhar和Moun确定了gr^3_2(B_3)=6、gr^3_2(B_4)=7。本文将其结果扩展至t>3种颜色的情况，确定了5≤n≤15时gr^3_2(B_n)的值，还给出了gr^t_2(B_n)的一般下界。
英文摘要
For $1\le s<t$ and any graph $G$, the weakened Gallai-Ramsey number $gr^t_s(G)$ is defined to be the least $p\in \mathbb{N}$ such that every Gallai $t$-coloring of the edges of $K_p$ (i.e., a $t$-coloring that lacks rainbow triangles) contains a subgraph isomorphic to $G$ whose edges use at most $s$ of the colors. In the case of a book graph $B_n:=K_2+nK_1$, Jakhar and Moun determined the values $gr^3_2(B_3)=6$ and $gr^3_2(B_4)=7$. In this paper, we extend their results to $t>3$ colors, and we determine the values of $gr^3_2(B_n)$ for $5\le n\le 15$. General lower bounds for $gr^t_2(B_n)$ are also given.
Comments32 pages, 10 figures
