Ramsey goodness of stars and fans for the Haj\'os graph

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 $\chi$ 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 $n \ge s$, \[R(G_1,G_2)=(\chi-1)(n-1)+s.\] The $G_1$-good property has been extensively studied for star-like graphs when $G_1$ is a graph with $\chi(G_1)\ge 3$, 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 Haj\'os 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.