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|a 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 ≥ 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 1st statement is a conjecture of Thomassen, the 2nd is a conjecture of Dean, the 3rd is a tight answer to a question of Bondy and Vince, and the 4th is a conjecture of Sudakov and Verstraëte. All of the above results are best possible. © 2021 The Author(s).
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