A unified proof of conjectures on cycle lengths in graphs

In this paper, we prove a tight minimum degree condition in general graphs for the existence of paths between two given endpoints, whose lengths form a long arithmetic progression with common difference one or two. This allows us to obtain a number of exact and optimal results on cycle lengths in graphs of given minimum degree, connectivity or chromatic number. More precisely, we prove the following statements by a unified approach. (1) Every graph $G$ with minimum degree at least $k+1$ contains cycles of all even lengths modulo $k$; in addition, if $G$ is 2-connected and non-bipartite, then it contains cycles of all lengths modulo $k$. (2) For all $k\geq 3$, every $k$-connected graph contains a cycle of length zero modulo $k$. (3) Every 3-connected non-bipartite graph with minimum degree at least $k+1$ contains $k$ cycles of consecutive lengths. (4) Every graph with chromatic number at least $k+2$ contains $k$ cycles of consecutive lengths. The first statement is a conjecture of Thomassen, the second is a conjecture of Dean, the third is a tight answer to a question of Bondy and Vince, and the fourth is a conjecture of Sudakov and Verstra\"ete. All of the above results are best possible.