Colourings of graphs with two consecutive odd cycle lengths

In 1992 Gyárfás showed that a graph G having only k odd cycle lengths is ( 2 k + 1 ) -colourable, if it does not contain a K 2 k + 2 . In this paper, we will present the results for graphs containing only odd cycles of length 2 m − 1 and 2 m + 1 as done in [S. Matos Camacho, Colourings of graph with prescribed cycle lengths, diploma thesis, TU Bergakademie Freiberg, 2006. [3]]. We will show that these graphs are 4-colourable.