Extending Perfect Matchings to Hamiltonian Cycles in Line Graphs

Abstract

A graph admitting a perfect matching has the Perfect–Matching–Hamiltonian property (for short the PMH–property) if each of its perfect matchings can be extended to a hamiltonian cycle. In this paper we establish some sufficient conditions for a graph $G$ in order to guarantee that its line graph $L(G)$ has the PMH–property. In particular, we prove that this happens when $G$ is (i) a Hamiltonian graph with maximum degree at most 3, (ii) a complete graph, (iii) a balanced complete bipartite graph with at least 100 vertices, or (iv) an arbitrarily traceable graph. Further related questions and open problems are proposed along the paper.