An approach to the girth problem in cubic graphs

Abstract

We offer a new, gradual approach to the largest girth problem for cubic graphs. It is easily observed that the largest possible girth of all $n$-vertex cubic graphs is attained by a $2$-connected graph $G=(V,E)$. By Petersen's graph theorem, $E$ is the disjoint union of a $2$-factor and a perfect matching $M$. We refer to the edges of $M$ as chords and classify the cycles in $G$ by their number of chords. We define ${\hat{I}}_k^3(n)$ to be the largest integer $g$ such that every cubic $n$-vertex graph with a given perfect matching $M$ has a cycle of length at most $g$ with at most $k$ chords. Here we determine this function up to small additive constant for $k=1,2$ and up to a small multiplicative constant for larger $k$.