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.