On graphs without cycles of length 0 modulo 4

Abstract

Bollobás proved that for every k and ℓ such that $kZ+\ell$ contains an even number, an n-vertex graph containing no cycle of length $\ell \mathrm{mod}k$ can contain at most a linear number of edges. The precise (or asymptotic) value of the maximum number of edges in such a graph is known for very few pairs ℓ and k. In this work we precisely determine the maximum number of edges in a graph containing no cycle of length $0\mathrm{mod}4$.