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, or (iii) an arbitrarily traceable graph. Further related questions and open problems are proposed along the paper.
Extending perfect matchings to Hamiltonian cycles in line graphs / Abreu, M; Gauci, Jb; Labbate, D; Mazzuoccolo, G; Zerafa, Jp. - In: ELECTRONIC JOURNAL OF COMBINATORICS. - ISSN 1077-8926. - 28:1(2021), pp. 1-13. [10.37236/9143]
Extending perfect matchings to Hamiltonian cycles in line graphs
Mazzuoccolo, G;
2021
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, or (iii) an arbitrarily traceable graph. Further related questions and open problems are proposed along the paper.
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