An approach to the girth problem in cubic graphs

TL;DR

This paper introduces a new approach to the girth problem in cubic graphs by classifying cycles based on chords and determining bounds for the largest girth with limited chords.

Contribution

It defines the function  to analyze girth bounds in cubic graphs with perfect matchings, providing bounds for small and larger numbers of chords.

Findings

Determined  for k=1,2 with small additive constants.

Bounded  for larger k with a small multiplicative constant.

Provided a gradual approach to the largest 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 $\gamma_k(n)$ 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 $k$.