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 K₊+₁ or K₊, ₍-₊; (2) G contains cycles of lengths modulo k for all even, unless G K₊+₁ or K₊, ₍-₊; (3) G contains cycles of lengths modulo k for all, unless G K₊+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.
Bai et al. (Wed,) studied this question.