Minimum degree conditions for the existence of cycles of all lengths modulo $k$ in graphs

Shuya Chiba; Tomoki Yamashita

Minimum degree conditions for the existence of cycles of all lengths modulo $k$ in graphs

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arxiv 1904.03818 v1 pith:6BS2XKPS submitted 2019-04-08 math.CO

Shuya Chiba , Tomoki Yamashita This is my paper

classification math.CO

keywords cyclesdegreelengthsminimummoduloaffirmativelyconditionsconjecture

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abstract

Thomassen, in 1983, conjectured that for a positive integer $k$, every $2$-connected non-bipartite graph of minimum degree at least $k + 1$ contains cycles of all lengths modulo $k$. In this paper, we settle this conjecture affirmatively.

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A note on two cycles of consecutive even lengths in graphs
math.CO 2025-06 conditional novelty 6.0 of 10

If an n-vertex graph has at least 10q + binom(r+1,2) edges, where n-1 = 4q+r, then it contains two cycles of consecutive even lengths unless it is a chain of K5 blocks and one K_{r+1} block.

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