Induced 2-Regular Subgraphs in k-Chordal Cubic Graphs

TL;DR

This paper investigates the existence and size of induced 2-regular subgraphs in k-chordal cubic graphs, providing bounds based on cycle length restrictions and independence number, with structural insights into 4-chordal graphs.

Contribution

It establishes new lower bounds for induced 2-regular subgraphs in k-chordal cubic graphs and characterizes cubic 4-chordal graphs structurally.

Findings

Lower bound of (n-2)/(4 - 4/k) for graphs with no long induced cycles

Bound of (5n+6)/8 for 4-chordal cubic graphs with n>6

Induced 2-regular subgraph of at least (1/4 + ε)n when independence number is small

Abstract

We show that a cubic graph $G$ of order $n$ has an induced $2$-regular subgraph of order at least a) $\frac{n-2}{4-\frac{4}{k}}$, if $G$ has no induced cycle of length more than $k$, b) $\frac{5n+6}{8}$, if $G$ has no induced cycle of length more than $4$, and $n>6$, and c) $\left(\frac{1}{4}+\epsilon\right)n$, if the independence number of $G$ is at most $\left(\frac{3}{8}-\epsilon\right)n$. To show the second result we give a precise structural description of cubic $4$-chordal graphs.