Disproofs of four Gallai-Ramsey-type conjectures

TL;DR

This paper disproves three conjectures on Gallai-Ramsey numbers related to fans, wheels, and kipases by establishing new lower bounds, and also refutes a conjecture linking Ramsey-full and Gallai-Ramsey-full properties of graphs.

Contribution

It provides counterexamples and new bounds that disprove existing conjectures on Gallai-Ramsey numbers and graph properties.

Findings

Disproved three Gallai-Ramsey conjectures using new lower bounds.

Identified graphs that are Ramsey-full but not Gallai-Ramsey-full.

Refuted the conjecture linking Ramsey-full and Gallai-Ramsey-full graph classes.

Abstract

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.