Cycle lengths in graphs of given minimum degree

In a graph, k cycles are { admissible} if their lengths form an arithmetic progression with common difference one or two. Let G be a 2-connected graph with minimum degree at least k 4. We prove that {itemize} [(1)] G contains k admissible cycles, unless G Kₖ₊₁ or Kk,n-k; [(2)] G contains cycles of lengths modulo k for all even , unless G Kₖ₊₁ or Kk,n-k; [(3)] G contains cycles of lengths modulo k for all , unless G Kₖ₊₁ or G is bipartite. {itemize} In addition, we show that if k is even and G is 2-connected with minimum degree at least $k-1$ and order at least $k+2$, then G contains cycles of lengths modulo k for all even . These findings provide a stability analysis of the main results on cycle lengths in graphs of given minimum degree in [J. Gao, Q. Huo, C. Liu, J. Ma, A unified proof of conjectures on cycle lengths in graphs, International Mathematics Research Notices 2022 (10) (2022) 7615--7653]. As a corollary, we determine the maximum number of edges in a graph that does not contain a cycle of length 0 modulo k for all odd k.