Maximum Cardinality 1-Restricted Simple 2-Matchings

A simple 2-matching in a graph is a subgraph all of whose nodes have degree $1$ or $2$. A simple 2-matching is called $k$ -restricted if every connected component has $>k$ edges. We consider the problem of finding a $k$-restricted simple 2-matching with a maximum number of edges, which is a relaxation of the problem of finding a Hamilton cycle in a graph. Our main result is a min-max theorem for the maximum number of edges in a 1-restricted simple 2-matching. We prove this result constructively by presenting a polynomial time algorithm for finding a 1-restricted simple 2-matching with a maximum number of edges.