Every 2-connected n-vertex graph with more than floor((3n-1)/2) edges contains a cycle whose length is a multiple of 4, and this bound is sharp for all n at least 12.
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On $2$-connected graphs avoiding cycles of length $0$ modulo $4$
Every 2-connected n-vertex graph with more than floor((3n-1)/2) edges contains a cycle whose length is a multiple of 4, and this bound is sharp for all n at least 12.