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.
The rook graph is a graph whose edges represent all the possible legal moves of the rook chess piece on a chessboard. The problem we consider is the following. Given any set M containing pairs of cells such that each cell of the m 1 × m 2 chessboard is in exactly one pair, we determine the values of the positive integers m 1 and m 2 for which it is possible to construct a closed tour of all the cells of the chessboard which uses all the pairs of cells in M and some edges of the rook graph. This is an alternative formulation of a graphtheoretical problem presented in [Electron. J. Combin. 28(1) (2021), #P1.7] involving the Cartesian product G of two complete graphs K m1 and K m2 , which is, in fact, isomorphic to the m 1 × m 2 rook graph. The problem revolves around determining the values of the parameters m 1 and m 2 that would allow any perfect matching of the complete graph on the same vertex set of G to be extended to a Hamiltonian cycle by using only edges in G.
In 1994, it was conjectured by Fan and Raspaud that every simple bridgeless cubic graph has three perfect matchings whose intersection is empty. In this paper we answer a question recently proposed by Mkrtchyan and Vardanyan, by giving an equivalent formulation of the Fan-Raspaud Conjecture. We also study a possibly weaker conjecture originally proposed by the first author, which states that in every simple bridgeless cubic graph there exist two perfect matchings such that the complement of their union is a bipartite graph. Here, we show that this conjecture can be equivalently stated using a variant of Petersen-colourings, we prove it for graphs having oddness at most four and we give a natural extension to bridgeless cubic multigraphs and to certain cubic graphs having bridges.
A graph G has the Perfect-Matching-Hamiltonian property (PMH-property) if for each one of its perfect matchings, there is another perfect matching of G such that the union of the two perfect matchings yields a Hamiltonian cycle of G. The study of graphs that have the PMH-property, initiated in the 1970s by Las Vergnas and Häggkvist, combines three well-studied properties of graphs, namely matchings, Hamiltonicity and edge-colourings. In this work, we study these concepts for cubic graphs in an attempt to characterise those cubic graphs for which every perfect matching corresponds to one of the colours of a proper 3-edge-colouring of the graph. We discuss that this is equivalent to saying that such graphs are even-2-factorable (E2F), that is, all 2-factors of the graph contain only even cycles. The case for bipartite cubic graphs is trivial, since if G is bipartite then it is E2F. Thus, we restrict our attention to non-bipartite cubic graphs. A sufficient, but not necessary, condition for a cubic graph to be E2F is that it has the PMH-property.
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