Every n-vertex graph with no cycle of length 0 mod 3 or 4 mod 6 has at most (11/8)n - 7/4 edges, and equality occurs exactly for the constructed graphs H_k.
Bollob´ as, Cycles modulok,Bulletin of the London Mathematical Society9 (1) (1977), 97–98
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On graphs without cycles of length $0$ modulo $3$ or $4$ modulo $6$
Every n-vertex graph with no cycle of length 0 mod 3 or 4 mod 6 has at most (11/8)n - 7/4 edges, and equality occurs exactly for the constructed graphs H_k.