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

TL;DR

This paper investigates the Ramsey goodness of star and fan graphs relative to the Hajós graph, extending previous work to graphs with chromatic surplus 2 and establishing new conditions for their Ramsey numbers.

Contribution

It extends the study of Ramsey goodness to graphs with chromatic surplus 2, specifically analyzing the Hajós graph and providing new results for stars and fans.

Findings

Star $K_{1,n}$ is $H_a$-good if and only if $n$ is even.

Fan $F_n$ with $n extgreater 111$ is $H_a$-good.

Provides new conditions for Ramsey numbers involving Hajós graphs.

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.