The chromatic number of triangle-free and broom-free graphs in terms of the number of vertices

Abstract

The Gárfás–Sumner conjecture asks whether for every tree T, the class of (induced) T-free graphs is $\chi$-bounded. The conjecture is solved for several special trees, but it is still open in general. Motivated by the conjecture, the chromatic number of triangle-free and broom-free graphs is well studied, since a broom is one of the generalizations of a star, where a broom B(m, n) is the graph obtained from a star $K_{1,n}$ and an m-vertex path $P_m$ by identifying the center of $K_{1,n}$ and a leaf of $P_m$. Gárfás, Szemeredi and Tuza proved that for every triangle-free and B(m, n)-free graph G, $\chi (G) \le m+n-1$. This upper bound has been improved by Wang and Wu to $m+n-2$ for $m\ge 2, n\ge 1$. In this paper, we prove that any triangle-free and B(4, 2)-free graph G is 3-colorable if the number of vertices of G is at least 17. Furthermore, the above estimation is the best possible since there exists a triangle-free and B(4, 2)-free 4-chromatic graph with sixteen vertices, named the Clebsch graph.