2 research outputs found
Betwixt and between 2-factor Hamiltonian and perfect-matching-Hamiltonian graphs
A Hamiltonian graph is 2-factor Hamiltonian (2FH) if each of its 2-factors is
a Hamiltonian cycle. A similar, but weaker, property is the Perfect-Matching Hamiltonian property (PMH-property): a graph admitting a perfect matching is
said to have this property if each one of its perfect matchings (1-factors) can be
extended to a Hamiltonian cycle. It was shown that the star product operation
between two bipartite 2FH-graphs is necessary and sufficient for a bipartite graph
admitting a 3-edge-cut to be 2FH. The same cannot be said when dealing with the
PMH-property, and in this work we discuss how one can use star products to obtain
graphs (which are not necessarily bipartite, regular and 2FH) admitting the PMH property with the help of malleable vertices, which we introduce here. We show that the presence of a malleable vertex in a graph implies that the graph has the
PMH-property, but does not necessarily imply that it is 2FH. It was also conjectured
that if a graph is a bipartite cubic 2FH-graph, then it can only be obtained from
the complete bipartite graph K3,3 and the Heawood graph by using star products.
Here, we show that a cubic graph (not necessarily bipartite) is 2FH if and only if all
of its vertices are malleable. We also prove that the above conjecture is equivalent
to saying that, apart from the Heawood graph, every bipartite cyclically 4-edge connected cubic graph with girth at least 6 having the PMH-property admits a
perfect matching which can be extended to a Hamiltonian cycle in exactly one way.
Finally, we also give two necessary and sufficient conditions for a graph admitting
a 2-edge-cut to be: (i) 2FH, and (ii) PMH.peer-reviewe
On 3-Cut Reductions of Minimally 1-Factorable Cubic Bigraphs
AbstractA cubic bigraph G is minimally 1-factorable if every 1-factor lies in precisely one 1-factorization. We characterize 3-bridges of G and prove that the 3-cut reductions of G are still minimally 1-factorable; thus, the open classification problem is reduced to the study of 3-bridge-free minimally 1-factorable cubic bigraphs. Furthermore, we prove that if ab, cd are edges of G such that the graph obtained by twisting them in ad, bc is still minimally 1-factorable, then ab, cd lie in some 3-bridge of G