Cycles with consecutive odd lengths

In this paper we prove that there exists an absolute constant c>0c>0 such that for every natural number kk, every non-bipartite 2-connected graph with average degree at least ckck contains kk cycles with consecutive odd lengths. This implies the existence of the absolute constant d>0d>0 that every non-bipartite 2-connected graph with minimum degree at least dkdk contains cycles of all lengths modulo kk, thus providing an answer (in a strong form) to a question of Thomassen in Thomassen (1983). Both results are sharp up to the constant factors.