# Largest 2-regular subgraphs in 3-regular graphs

## TL;DR

This paper determines the minimum size of the largest 2-regular subgraph in 3-regular graphs, providing sharp bounds and characterizing extremal multigraphs.

## Contribution

It establishes exact bounds for the largest 2-regular subgraph in 3-regular and maximum degree 3 multigraphs, extending previous results and characterizing extremal cases.

## Key findings

- Bounds are sharp and optimal.
- Every 3-regular multigraph with c cut-edges has a nearly spanning 2-regular subgraph.
- General bounds apply to multigraphs with maximum degree 3 and given edge counts.

## Abstract

For a graph $G$, let $f_2(G)$ denote the largest number of vertices in a $2$-regular subgraph of $G$. We determine the minimum of $f_2(G)$ over $3$-regular $n$-vertex simple graphs $G$. To do this, we prove that every $3$-regular multigraph with exactly $c$ cut-edges has a $2$-regular subgraph that omits at most $\max\{0,\lfloor (c-1)/2\rfloor\}$ vertices. More generally, every $n$-vertex multigraph with maximum degree $3$ and $m$ edges has a $2$-regular subgraph that omits at most $\max\{0,\lfloor (3n-2m+c-1)/2\rfloor\}$ vertices. These bounds are sharp; we describe the extremal multigraphs.