Gallai-Ramsey Numbers for $\ell$-Connected Graphs

TL;DR

The paper studies Gallai-Ramsey numbers $gr_k(G:\,F^\ell)$ for $G$ in {P5, K_{1,3}} and the family $F^\ell$ of $\ell$-connected graphs. It leverages structural results for rainbow $P_5$-free and rainbow $K_{1,3}$-free colorings to obtain exact values and tight bounds across regimes of $k$ and $\ell$, including several closed-form thresholds. For $G= P_5$, it provides exact values for $\mathcal{F}^2$ and $\mathcal{F}^3$, precise results for $\mathcal{F}^4$, and general bounds plus exact $gr_k(P_5:\,F_n^{\ell})$ in ranges such as $n\ge \ell+3$, $2\le \ell \le 2k-4$. For $G= K_{1,3}$, it establishes analogous thresholds, delivering an exact formula $gr_k(K_{1,3}:F^\ell)=ceil((1+sqrt(1+8k))/2)$ when $k\ge ceil(\ell/2)+2$ and tight bounds when $k$ is smaller. These results advance the understanding of Gallai-Ramsey thresholds under connectivity constraints in multicolor Ramsey theory.

Abstract

Given a nonempty graph $G$, a collection of nonempty graphs $\cal{H}$, and a positive integer $k$, the Gallai-Ramsey number $\mathrm{gr}_k(G:\mathcal{H})$ is defined to be the minimum positive integer $n$ such that every exact $k$-edge-coloring of a complete graph $K_n$ contains either a rainbow copy of $G$ or a monochromatic copy of some element in $\mathcal{H}$. In this paper, we obtain some exact values and general lower and upper bounds for $\mathrm{gr}_k(G:\mathcal{F}^\ell)$, where $\mathcal{F}^\ell$ is the set of $\ell$-connected graphs and $G\in\{P_5, K_{1,3}\}$.

Theorems & Definitions (17)

• Theorem 1.1: Ma83
• Definition 1
• Theorem 1.2: TW07
• Theorem 1.3: BMOPGLST87
• Theorem 2.1: wei
• Theorem 2.2
• proof
• Theorem 2.3
• proof
• Theorem 2.4