Ramsey numbers and Gallai--Ramsey numbers of disjoint unions of cherries

TL;DR

This paper investigates Ramsey and Gallai--Ramsey numbers for disjoint unions of cherries, disproves a conjecture for Ramsey numbers, and confirms it for Gallai--Ramsey numbers, providing new insights into their exact values.

Contribution

It disproves a conjecture on Ramsey numbers for disjoint unions of cherries and confirms the Gallai--Ramsey conjecture, offering conditions for exact Ramsey number calculations.

Findings

Disproved the proposed Ramsey number conjecture for disjoint unions of cherries.

Confirmed the Gallai--Ramsey conjecture for the same class of graphs.

Provided sufficient conditions for determining exact Ramsey numbers.

Abstract

For graphs $G_1,\ldots,G_k$, the Ramsey number $R(G_1,\ldots,G_k)$ is the smallest positive integer $N$ such that every $k$-edge-coloring of $K_N$ contains a monochromatic copy of $G_i$ in color $i$ for some $i\in[k]$. The Gallai--Ramsey number $GR(G_1,\ldots,G_k)$ is defined analogously, with the colorings restricted to Gallai colorings (i.e., edge-colorings with no rainbow triangle). A copy of $P_3$ is called a cherry. Let $n_iP_3$ denote the disjoint union of $n_i$ cherries. Wu, Magnant, Nowbandegani, and Xia (Discrete Appl. Math., 2019) proposed two conjectures: \[ R(n_1P_3,\ldots,n_kP_3)=N\ \text{and}\ GR(n_1P_3,\ldots,n_kP_3)=N\,, \] where $N=2\max\{n_1,\ldots,n_k\}+\sum_{i=1}^kn_i-k+1$. We disprove the Ramsey conjecture and provide some sufficient conditions for determining the exact value of $R(n_1P_3,\ldots,n_kP_3)$. In contrast, we confirm the Gallai--Ramsey conjecture.