Cycle lengths and chromatic number of graphs Original Research Article

For a simple finite graph G let Co(G) and Ce(G) denote the set of odd cycle lengths and even cycle lengths in a graph G, respectively. We will show that the chromatic number χ(G) of G satisfies: χ(G)⩽ min{2r+2,2s+3}⩽r+s+2, if |Co(G)|=r and |Ce(G)|=s.