The Ramsey Number for a Forest versus Disjoint Union of Complete Graphs

TL;DR

This paper confirms the exact Ramsey number for any tree versus a disjoint union of two complete graphs, extending understanding of graph coloring and Ramsey theory for these specific graph classes.

Contribution

It establishes the Ramsey number for trees against disjoint unions of complete graphs, showing trees are $K_m up K_l$-good, which was previously unknown.

Findings

Exact Ramsey number for trees versus disjoint unions of complete graphs.

Trees are shown to be $K_m up K_l$-good.

Extends Ramsey theory for specific graph classes.

Abstract

Given two graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the minimum integer $N$ such that any coloring of the edges of $K_N$ in red or blue yields a red $G$ or a blue $H$. Let $v(G)$ be the number of vertices of $G$ and $\chi(G)$ be the chromatic number of $G$. Let $s(G)$ denote the chromatic surplus of $G$, the cardinality of a minimum color class taken over all proper colorings of $G$ with $\chi(G)$ colors. Burr showed that for a connected graph $G$ and a graph $H$ with $v(G)\geq s(H)$, $R(G,H) \geq (v(G)-1)(\chi(H)-1)+s(H)$. A connected graph $G$ is called $H$-good if $R(G,H)=(v(G)-1)(\chi(H)-1)+s(H)$. In this paper, we mainly confirm the Ramsey number for any tree $T_n$ versus $K_m\cup K_l$. Our result yields that $T_n$ is $K_m\cup K_l$-good.