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In this paper, we prove a tight minimum degree condition in general graphsfor the existence of paths between two given endpoints, whose lengths form along arithmetic progression with common difference one or two. This allows usto obtain a number of exact and optimal results on cycle lengths in graphs ofgiven 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 evenlengths modulo $k$; in addition, if $G$ is 2-connected and non-bipartite, thenit 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) Every3-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 aconjecture of Thomassen, the second is a conjecture of Dean, the third is atight answer to a question of Bondy and Vince, and the fourth is a conjectureof Sudakov and Verstra\"ete. All of the above results are best possible.

A unified proof of conjectures on cycle lengths in graphs