Ramsey goodness of stars and fans for the Hajós graph
Jiafu He, Haiyu Zeng, Yanbo Zhang
TL;DR
This work determines Ramsey-goodness for the Hajós graph $H_a$ (chromatic surplus 2) against two sparse targets. It proves $K_{1,n}$ is $H_a$-good iff $n$ is even, yielding $R(H_a,K_{1,n})=2n+2$ for even $n\ge2$ and $R(H_a,K_{1,n})=2n+3$ for odd $n\ge3$, and it proves $F_n$ is $H_a$-good for $n\ge111$, with $R(H_a,F_n)=4n+2$. The upper bounds rely on high-degree and subgraph-structure arguments, plus a crucial lemma $R(W_4,F_n)=4n+1$ for $n\ge111$, to control how the complement can avoid $F_n$. Overall, the paper extends Ramsey-goodness from surplus-1 to surplus-2 graphs, highlighting parity effects for stars and large-n behavior for fans in sparse Ramsey theory.
Abstract
Given two graphs $G_1$ and $G_2$, the Ramsey number $R(G_1,G_2)$ denotes the smallest integer $N$ such that any red-blue coloring of the edges of $K_N$ contains either a red $G_1$ or a blue $G_2$. Let $G_1$ be a graph with chromatic number $χ$ and chromatic surplus $s$, and let $G_2$ be a connected graph with $n$ vertices. The graph $G_2$ is said to be Ramsey-good for the graph $G_1$ (or simply $G_1$-good) if, for $n \ge s$, \[R(G_1,G_2)=(χ-1)(n-1)+s.\] The $G_1$-good property has been extensively studied for star-like graphs when $G_1$ is a graph with $χ(G_1)\ge 3$, as seen in works by Burr-Faudree-Rousseau-Schelp (J. Graph Theory, 1983), Li-Rousseau (J. Graph Theory, 1996), Lin-Li-Dong (European J. Combin., 2010), Fox-He-Wigderson (Adv. Combin., 2023), and Liu-Li (J. Graph Theory, 2025), among others. However, all prior results require $G_1$ to have chromatic surplus $1$. In this paper, we extend this investigation to graphs with chromatic surplus 2 by considering the Hajós graph $H_a$. For a star $K_{1,n}$, we prove that $K_{1,n}$ is $H_a$-good if and only if $n$ is even. For a fan $F_n$ with $n\ge 111$, we prove that $F_n$ is $H_a$-good.