Ramsey numbers of sparse graphs versus disjoint books

TL;DR

This paper determines the Ramsey number for a connected sparse graph versus multiple disjoint books, extending previous results from trees to more general sparse graphs with a specific edge bound.

Contribution

It generalizes the known Ramsey number results from trees to connected graphs with limited edges, providing an explicit formula for large enough graphs.

Findings

Ramsey number for connected sparse graphs versus disjoint books is 2n + t - 2.

Result holds for graphs with at most n(1+ε) edges, where ε depends on k and t.

Established for sufficiently large n, specifically n ≥ 111t^3k^3.

Abstract

Let $B_k$ denote a book on $k+2$ vertices and $tB_k$ be $t$ vertex-disjoint $B_k$'s. Let $G$ be a connected graph with $n$ vertices and at most $n(1+\epsilon)$ edges, where $\epsilon$ is a constant depending on $k$ and $t$. In this paper, we show that the Ramsey number $$r(G,tB_k)=2n+t-2$$ provided $n\ge 111t^3k^3$. Our result extends the work of Erd\H{o}s, Faudree, Rousseau, and Schelp (1988), who established the corresponding result for $G$ being a tree and $t=1$.