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