标题： 四条Gallai-Ramsey型猜想的反例 显示英文标题

标题： Disproofs of four Gallai-Ramsey-type conjectures

摘要： 作为Ramsey数的一个重要变种，Gallai-Ramsey数$GR_k(H)$指的是最小的正整数$r$，使得通过用至多$k$种颜色对$K_r$的边进行着色，存在一个与$H$同构的单色子图或者一个彩虹三角形。 Mao、Wang、Magnant和Schiermeyer [离散数学，2023]，Song、Wei、Zhang和Zhao [离散数学，2020]，以及Zhao和Wei [离散应用数学，2021] 分别提出了关于扇形图、轮图和kipases的Gallai-Ramsey数的三个猜想。 我们建立了新的下界，从而否定了这三个猜想。 Su和Liu [图论组合学，2022] 研究了图的Gallai-Ramsey全性质，并猜想一个图是Ramsey全当且仅当它是Gallai-Ramsey全。 我们给出了两类Ramsey全的图，但它们都不是Gallai-Ramsey全。 显示英文摘要

摘要： As a significant variation of Ramsey numbers, the Gallai-Ramsey number $GR_k(H)$ refers to the smallest positive integer $r$ such that, by coloring the edges of $K_r$ with at most $k$ colors, there exists either a monochromatic subgraph isomorphic to $H$ or a rainbow triangle. Mao, Wang, Magnant, and Schiermeyer [Discrete Math., 2023], Song, Wei, Zhang, and Zhao [Discrete Math., 2020], and Zhao and Wei [Discrete Appl. Math., 2021] each proposed one conjecture on the Gallai-Ramsey numbers for fans, wheels, and kipases, respectively. We establish new lower bounds that disprove all three conjectures. Su and Liu [Graphs Combin., 2022] studied the Gallai-Ramsey-full property of graphs and conjectured that a graph is Ramsey-full if and only if it is Gallai-Ramsey-full. We present two classes of graphs that are Ramsey-full, but neither is Gallai-Ramsey-full.