Trichotomy and $tK_m$-goodness of sparse graphs

TL;DR

This paper establishes a new trichotomy for sparse graphs, showing under certain conditions that the Ramsey number involving these graphs and disjoint unions of complete graphs is exactly determined, extending previous results.

Contribution

It introduces a trichotomy framework for sparse graphs and proves the exact Ramsey number for a broad class of such graphs, generalizing earlier specific cases.

Findings

For large n, the Ramsey number r(G, tK_m) equals (n-1)(m-1)+t.

The result applies when 1 ≤ k ≤ c n^{2/(m-1)} for some constant c.

Extends known cases for t=1 and k=1 to more general sparse graphs.

Abstract

Let $G$ be a connected graph with $n$ vertices and $n+k-2$ edges and $tK_m$ denote the disjoint union of $t$ complete graphs $K_m$. In this paper, by developing a trichotomy for sparse graphs, we show that for given integers $m\ge 2$ and $t\ge 1$, there exists a positive constant $c$ such that if $1\le k\le cn^{\frac{2}{m-1}}$ and $n$ is large, then $G$ is $tK_m$-good, that is, the Ramsey number is \[ r(G, tK_m)=(n-1)(m-1)+t\,. \] In particular, the above equality holds for any positive integers $k$, $m$, and $t$, provided $n$ is large. The case $t=1$ was obtained by Burr, Erd\H{o}s, Faudree, Rousseau, and Schelp (1980), and the case $k=1$ was established by Luo and Peng (2023).