Colourings of graphs with two consecutive odd cycle lengths

In 1992 Gyárfás showed that a graph G having only kk odd cycle lengths is (2k+1)(2k+1)-colourable, if it does not contain a K2k+2K2k+2. In this paper, we will present the results for graphs containing only odd cycles of length 2m−12m−1 and 2m+12m+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.