Upper bounds on the chromatic number of triangle-free graphs with a forbidden subtree

Abstract

Gyárfás conjectured that for a given forest F, there exists an integer function f(F, x) such that $\chi (G)\le f(F,\omega (G))$ for each F-free graph G, where $\omega (G)$ is the clique number of G. The broom B(m, n) is the tree of order $m+n$ obtained from identifying a vertex of degree 1 of the path $P_m$ with the center of the star $K_{1,n}$. In this note, we prove that every connected, triangle-free and B(m, n)-free graph is $(m+n-2)$-colorable as an extension of a result of Randerath and Schiermeyer and a result of Gyárfás, Szemeredi and Tuza. In addition, it is also shown that every connected, triangle-free, $C_4$-free and T-free graph is $(p-2)$-colorable, where T is a tree of order $p\ge 4$ and $T\not \cong K_{1,3}$.