Largest 2-Regular Subgraphs in 3-Regular Graphs

Abstract

For a graph G, let f2(G) denote the largest number of vertices in a 2-regular subgraph of G. We determine the minimum of f2(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,(c-1)/2} 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,(3n-2m+c-1)/2} vertices. These bounds are sharp; we describe the extremal multigraphs.