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, ⌊(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.Mathematics Subject Classification: 05C07, 05C70, 05C35.