Fan-goodness of sparse graphs

TL;DR

This paper investigates the Ramsey numbers involving sparse graphs and fans, establishing bounds based on the number of edges and extending previous results to more general sparse graph classes.

Contribution

It generalizes fan-goodness results to broader sparse graphs, providing new bounds on Ramsey numbers depending on edge counts and parameters.

Findings

If a graph has at most $n(1+ ext{constant})$ edges, its Ramsey number with a fan is $2n-1$.

For graphs with limited edges, the Ramsey number with multiple fans is $2n + t - 2$.

Results hold for sufficiently large $n$, specifically $n o ext{large}$ depending on parameters.

Abstract

Let $G$ be a connected graph of order $n$, $F_k$ be a fan consisting of $k$ triangles sharing a common vertex, and $tF_k$ be $t$ vertex-disjoint copies of $F_k$. Brennan (2017) showed the Ramsey number $r(G,F_k)=2n-1$ for $G$ being a unicyclic graph for $n \geq k^2-k+1$ and $k\ge 18$, and asked the threshold $c(n)$ for which $r(G,F_k) \geq 2n$ holds for any $G$ containing at least $c(n)$ cycles and $n$ being large. In this paper, we consider fan-goodness of general sparse graphs and show that if $G$ has at most $n(1+\epsilon(k))$ edges, where $\epsilon(k)$ is a constant depending on $k$, then $$r(G,F_k)=2n-1$$ for $n\ge 36k^4$, which implies that $c(n)$ is greater than $\epsilon(k) n$. Moreover, if $G$ has at most $n(1+\epsilon(k,t))$ edges, where $\epsilon(k,t)$ is a constant depending on $k,t$, then $$r(G,tF_k)=2n+t-2$$ provided $n\ge 161t^2k^4$.