  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.
