On graphs without cycles of length 1 modulo 3

Burr and Erdős conjectured in 1976 that for two integers $k>\ell\geqslant 0$ satisfying that $k\mathbb{Z}+\ell$ contains some even integer, an $n$-vertex graph containing no cycles of length $\ell$ mod $k$ can contain at most a linear number of edges on $n$. Bollobás confirmed this conjecture in 1977 and then Erdős proposed the problem of determining the exact value of the maximum number of edges in such a graph. For the above $k$ and $\ell$, define $c_{\ell,k}$ to be the least constant such that every $n$-vertex graph with at least $c_{\ell,k}\cdot n$ edges contains a cycle of length $\ell$ mod $k$. The precise (or asymptotic) values of $c_{\ell,k}$ are known for very few pairs $\ell$ and $k$. We precisely determine the maximum number of edges in a graph containing no cycles of length 1 mod 3. In particular, we show that every $n$-vertex graph with at least $\frac{5}{3}(n-1)$ edges contains a cycle of length 1 mod 3, unless $9\,\mid\,(n-1)$ and each block of the graph is a Petersen graph. As a corollary, we get that $c_{1,3}=\frac{5}{3}$. This is the last remaining class modulo $k$ for $1\leqslant k\leq 4$.