On $2$-connected graphs avoiding cycles of length $0$ modulo $4$

Abstract

For two integers $k$ and $\ell$, an $(\ell \text{ mod }k)$-cycle means a cycle of length $m$ such that $m\equiv \ell (\mathrm{mod}k)$. In 1977, Bollobás proved a conjecture of Burr and Erdős by showing that if $\ell$ is even or $k$ is odd, then every $n$-vertex graph containing no $(\ell \text{ mod }k)$-cycles has at most a linear number of edges in terms of $n$. Since then, determining the exact extremal bounds for graphs without $(\ell \text{ mod }k)$-cycles has emerged as an interesting question in extremal graph theory, though the exact values are known only for a few integers $\ell$ and $k$. Recently, Győri, Li, Salia, Tompkins, Varga and Zhu proved that every $n$-vertex graph containing no $(0\text{ mod }4)$-cycles has at most $\lfloor \frac{19}{12}(n-1)\rfloor$ edges, and they provided extremal examples that reach the bound, all of which are not $2$-connected. In this paper, we show that a $2$-connected graph without $(0\text{ mod }4)$-cycles has at most $\lfloor \frac{3n-1}{2}\rfloor$ edges, and this bound is tight by presenting a method to construct infinitely many extremal examples.