On the chromatic number of (P5, dart)-free graphs

For a graph G, omega(G), chi(G) represent the clique number and the chromatic number of G, respectively. A hereditary family G of graphs is called chi-bounded with chi-binding function f if chi(G) <= f (omega(G)) for all G is an element of G. A result of Schiermeyer shows that the class of (P-5, dart)-free graphs has a chi-binding function f(omega) = omega(2) [L. Esperet, L. Lemoine, F. Maffray and G. Morel, The chromatic number of (P-5, K-4)-free graphs, Discrete Math. 313 (2013) 743-754]. In this paper, we prove that the class of (P-5, dart)-free graphs has a chi-binding function f(omega) = 3/4 omega(2). For the class of (P-5, C-5, dart)-free graphs, we give a chi-binding function f(omega) = ((omega+1)(2)).