Largest 2-regular subgraphs in 3-regular graphs

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.