On the star-critical Ramsey number of a forest versus complete graphs

TL;DR

This paper computes the exact Ramsey and star-critical Ramsey numbers for forests versus complete graphs, providing new formulas and tight bounds for related size Ramsey numbers.

Contribution

It introduces explicit calculations of the star-critical Ramsey number for forests against complete graphs, linking it to known Ramsey numbers of complete graphs.

Findings

Exact formulas for Ramsey numbers of forests versus complete graphs.

Precise computation of star-critical Ramsey numbers in this context.

Tight bounds for size Ramsey numbers of forests versus complete graphs.

Abstract

Let $G$ and $G_1, G_2, \ldots , G_t$ be given graphs. By $G\rightarrow (G_1, G_2, \ldots , G_t)$ we mean if the edges of $G$ are arbitrarily colored by $t$ colors, then for some $i$, $1\leq i\leq t$, the spanning subgraph of $G$ whose edges are colored with the $i$-th color, contains a copy of $G_i$. The Ramsey number $R(G_1, G_2, \ldots, G_t)$ is the smallest positive integer $n$ such that $K_n\rightarrow (G_1, G_2, \ldots , G_t)$ and the size Ramsey number $\hat{R}(G_1, G_2, \ldots , G_t)$ is defined as $\min\{|E(G)|:~G\rightarrow (G_1, G_2, \ldots , G_t)\}$. Also, for given graphs $G_1, G_2, \ldots , G_t$ with $r=R(G_1, G_2, \ldots , G_t)$, the star-critical Ramsey number $R_*(G_1, G_2, \ldots , G_t)$ is defined as $\min\{\delta(G):~G\subseteq K_r, ~G\rightarrow (G_1, G_2, \ldots , G_t)\}$. In this paper, the Ramsey number and also the star-critical Ramsey number of a forest versus any number of complete graphs will be computed exactly in terms of the Ramsey number of complete graphs. As a result, the computed star-critical Ramsey number is used to give a tight bound for the size Ramsey number of a forest versus a complete graph.