论文标题
彩色数字三的Gallai-Ramsey编号
Gallai-Ramsey numbers for graphs with chromatic number three
论文作者
论文摘要
给定图形$ h $和一个整数$ k \ ge1 $，Gallai-ramsey编号$ gr_k（h）$被定义为最小整数$ n $，因此每个$ k $ - 完整的图形$ k_n $的每个$ k $ - 包含rainbow（所有不同的彩色）triangle或单块triangle或$ h $ $ h $ $ h $。在本文中，我们研究了具有色度三个的图形的Gallai-ramsey数字，例如$ \ wideHat {k} _m $ for $ m \ ge2 $，其中$ \ widehat {k} _m $是$ m+m+m+m+m+1 $的kipas，从$ k_1 $＆k_1 $和$ p_m $的dertice和a Grapher comphertice获得的$ m+1 $ covertice，以及$ k_1 $和$ p_m $ $ p_m $ and a n eftertice and devertice and a Grapher and a Grapher and and aGrap所$ \ mathscr {h} $。我们首先研究了此类图的一般下限，并提出了$ gr_k（\ wideHat {k} _m）$的确切值的猜想。然后，我们给出一个统一的证明，以确定$ \ mathscr {h} $中许多图的Gallai-ramsey编号，并获得$ gr_k（\ wideHat {k} _4）$的确切值，$ k \ ge1 $。我们的结果不仅表明$ gr_k（\ wideHat {k} _m）$的猜想对于$ m \ le4 $是正确的，而且暗示着$ gr_k（h）$的几个结果，对于一些$ h \ in \ mathscr {h} $，这些结果在不同论文中是单独证明的。
Given a graph $H$ and an integer $k\ge1$, the Gallai-Ramsey number $GR_k(H)$ is defined to be the minimum integer $n$ such that every $k$-edge coloring of the complete graph $K_n$ contains either a rainbow (all different colored) triangle or a monochromatic copy of $H$. In this paper, we study Gallai-Ramsey numbers for graphs with chromatic number three such as $\widehat{K}_m$ for $m\ge2$, where $\widehat{K}_m$ is a kipas with $m+1$ vertices obtained from the join of $K_1$ and $P_m$, and a class of graphs with five vertices, denoted by $\mathscr{H}$. We first study the general lower bound of such graphs and propose a conjecture for the exact value of $GR_k(\widehat{K}_m)$. Then we give a unified proof to determine the Gallai-Ramsey numbers for many graphs in $\mathscr{H}$ and obtain the exact value of $GR_k(\widehat{K}_4)$ for $k\ge1$. Our outcomes not only indicate that the conjecture on $GR_k(\widehat{K}_m)$ is true for $m\le4$, but also imply several results on $GR_k(H)$ for some $H\in \mathscr{H}$ which are proved individually in different papers.
