Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
144 tokens/sec
GPT-4o
8 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Ramsey goodness of fans (2310.13204v1)

Published 20 Oct 2023 in math.CO

Abstract: Given two graphs $G_1$ and $G_2$, the Ramsey number $r(G_1,G_2)$ refers to the smallest positive integer $N$ such that any graph $G$ with $N$ vertices contains $G_1$ as a subgraph, or the complement of $G$ contains $G_2$ as a subgraph. A connected graph $H$ is said to be $p$-good if $r(K_p,H)=(p-1)(|H|-1)+1$. A generalized fan, denoted as $K_1+nH$, is formed by the disjoint union of $n$ copies of $H$ along with an additional vertex that is connected to each vertex of $nH$. Recently Chung and Lin proved that $K_1+nH$ is $p$-good for $n\ge cp\ell/|H|$, where $c\approx 52.456$ and $\ell=r(K_{p},H)$. They also posed the question of improving the lower bound of $n$ further so that $K_1+nH$ remains $p$-good. In this paper, we present three different methods to improve the range of $n$. First, we apply the Andr\'asfai-Erd\H{o}s-S\'os theorem to reduce $c$ from $52.456$ to $3$. Second, we utilize the approach established by Chen and Zhang to achieve a further reduction of $c$ to $2$. Lastly, we employ a new method to bring $c$ down to $1$. In addition, when $K_1+nH$ forms a fan graph $F_n$, we can further obtain a slightly more refined bound of $n$.

Summary

We haven't generated a summary for this paper yet.