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

Abstract

Gyarfas conjectured that for a given forest F, there exists an integer function f(F, x) such that $$\chi (G)\le f(F,\omega (G))$$ź(G)≤f(F,ź(G)) for each F-free graph G, where $$\omega (G)$$ź(G) is the clique number of G. The broom B(m, n) is the tree of order $$m+n$$m+n obtained from identifying a vertex of degree 1 of the path $$P_m$$Pm with the center of the star $$K_{1,n}$$K1,n. In this note, we prove that every connected, triangle-free and B(m, n)-free graph is $$(m+n-2)$$(m+n-2)-colorable as an extension of a result of Randerath and Schiermeyer and a result of Gyarfas, Szemeredi and Tuza. In addition, it is also shown that every connected, triangle-free, $$C_4$$C4-free and T-free graph is $$(p-2)$$(p-2)-colorable, where T is a tree of order $$p\ge 4$$pź4 and $$T\not \cong K_{1,3}$$TźK1,3.