On the Chromatic Number of ($$P_6$$, Diamond)-Free Graphs

In this paper, we first show that every (\(P_6\), diamond, \(K_4\))-free graph is 6-colorable. We also give an example of a (\(P_6\), diamond, \(K_4\))-free graph G with \(\chi (G)\) \( = 6\). Further, we show that for every (\(P_6\), diamond)-free graph G, the chromatic number of G is upper bounded by a linear function of its clique number. This generalizes some known results in the literature.