On chromatic number of graphs without certain induced subgraphs

Gyarfas conjectured that for a given forest F, there exists an integer function f (F, omega(G)) such that chi(G) <= f (F, omega(G)) for any F-free graph C, where chi(G) and omega(G) are, respectively, the chromatic number and the clique number of G. Let G be a C(5)-free graph and k be a positive integer. We show that if G is (kP(1) + P(2))-free for k >= 2, then chi(G) <= 2 omega(k-1) root omega; if G is (kP(1) + P(3))-free for k >= 1, then chi(G) <= omega(k) root omega. A graph G is k-divisible if for each induced subgraph H of G with at least one edge, there is a partition of the vertex set of H into k sets V(1), ... , V(k) such that no V(i) contains a clique of size omega(G). We show that a (2P(1)+P(2))-free and C(5)-free graph is 2-divisible.