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

TL;DR

This paper establishes a tight upper bound on the number of edges in 2-connected graphs avoiding cycles of length divisible by 4, extending previous results by providing extremal constructions.

Contribution

It proves a new extremal bound for 2-connected graphs avoiding $(0 ext{ mod } 4)$-cycles and introduces a method to construct infinitely many extremal examples.

Findings

Maximum edges in 2-connected $(0 ext{ mod } 4)$-cycle-free graphs is $loor{rac{3n-1}{2}}$.

Bound is tight with infinitely many extremal constructions.

Extends previous bounds for general graphs to the 2-connected case.

Abstract

For two integers $k$ and $\ell$, an $(\ell \text{ mod }k)$-cycle means a cycle of length $m$ such that $m\equiv \ell\pmod{k}$. In 1977, Bollob\'{a}s proved a conjecture of Burr and Erd\H{o}s 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, Gy\H{o}ri, Li, Salia, Tompkins, Varga and Zhu proved that every $n$-vertex graph containing no $(0 \text{ mod }4)$-cycles has at most $\left\lfloor \frac{19}{12}(n -1) \right\rfloor$ 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 $\left\lfloor \frac{3n-1}{2} \right\rfloor$ edges, and this bound is tight by presenting a method to construct infinitely many extremal examples.