Largest 2-Regular Subgraphs in 3-Regular Graphs

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 , ⌊ (3 n- 2 m+ c- 1) / 2 ⌈ } vertices. These bounds are sharp; we describe the extremal multigraphs.