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.
Bollob´ as, Cycles modulok, Bull
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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.