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Bollob\'as proved that for every $k$ and $\ell$ such that $k\mathbb{Z}+\ell$contains an even number, an $n$-vertex graph containing no cycle of length$\ell \bmod k$ can contain at most a linear number of edges. The precise (orasymptotic) value of the maximum number of edges in such a graph is known forvery few pairs $\ell$ and $k$. In this work we precisely determine the maximumnumber of edges in a graph containing no cycle of length $0 \bmod 4$.

On graphs without cycles of length 0 modulo 4