Summary: A. Gyárfás [Zastosow. Mat. 19, No. 3–4, 413–441 (1987; Zbl 0718.05041)] conjectured that for a given forest \(F\), there exists an integer function \(f(F,x)\) such that \(\chi (G)\leq f(F,\omega (G))\) 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 B. Randerath and I. Schiermeyer [Australas. J. Comb. 26, 3–9 (2002; Zbl 1018.05036)] and a result of A. Gyárfás et al. [Discrete Math. 30, 235–244 (1980; Zbl 0475.05027)]. 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 \(p\geq 4\) and \(T\not\cong K_{1,3}\).