COLORING GRAPHS WITH TWO ODD CYCLE LENGTHS

In this paper we determine the chromatic number of graphs with two odd cycle lengths. Let G be a graph and L(G) be the set of all odd cycle lengths of G. We prove that (1) if L(G) = {3, 3 + 2l}, where l >= 2, then chi(G) = max{3, omega(G)}, and (2) if L(G) = {k, k + 2l}, where k >= 5 and l >= 1, then chi(G) = 3. These, together with the case L(G) = {3, 5} solved in [S.-S. Wang, SIAM T. Discrete Math., 22 (2008), pp. 1040-1072] give a complete solution to the general problem addressed in [S.-S. Wang, SIAM T. Discrete Math., 22 (2008), pp. 1040-1072; S.-M. Camacho and I. Schiermeyer, Discrete Math., 309 (2009), pp. 4916-4919; and T. Kaiser, O .Rucky, and R. Skrekovski, SIAM T. Discrete Math., 25 (2011), pp. 1069-1088]. Our results also improve a classical theorem of Gyarfas which asserts that chi(G) <= 2 vertical bar L(G)vertical bar + 2 for any graph G.