7 research outputs found
Extending a perfect matching to a Hamiltonian cycle
Graph TheoryInternational audienceRuskey and Savage conjectured that in the d-dimensional hypercube, every matching M can be extended to a Hamiltonian cycle. Fink verified this for every perfect matching M, remarkably even if M contains external edges. We prove that this property also holds for sparse spanning regular subgraphs of the cubes: for every d ≥7 and every k, where 7 ≤k ≤d, the d-dimensional hypercube contains a k-regular spanning subgraph such that every perfect matching (possibly with external edges) can be extended to a Hamiltonian cycle. We do not know if this result can be extended to k=4,5,6. It cannot be extended to k=3. Indeed, there are only three 3-regular graphs such that every perfect matching (possibly with external edges) can be extended to a Hamiltonian cycle, namely the complete graph on 4 vertices, the complete bipartite 3-regular graph on 6 vertices and the 3-cube on 8 vertices. Also, we do not know if there are graphs of girth at least 5 with this matching-extendability property
Extending perfect matchings to Hamiltonian cycles in line graphs
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 in order to guarantee that its line graph has
the PMH-property. In particular, we prove that this happens when is (i) a
Hamiltonian graph with maximum degree at most , (ii) a complete graph, or
(iii) an arbitrarily traceable graph. Further related questions and open
problems are proposed along the paper.Comment: 12 pages, 4 figure
On a family of quartic graphs: Hamiltonicity, matchings and isomorphism with circulants
A pairing of a graph is a perfect matching of the underlying complete
graph . A graph has the PH-property if for each one of its pairings,
there exists a perfect matching of such that the union of the two gives
rise to a Hamiltonian cycle of . In 2015, Alahmadi et al. proved that the
only three cubic graphs having the PH-property are the complete graph ,
the complete bipartite graph , and the -dimensional cube
. Most naturally, the next step is to characterise the quartic
graphs that have the PH-property, and the same authors mention that there
exists an infinite family of quartic graphs (which are also circulant graphs)
having the PH-property. In this work we propose a class of quartic graphs on
two parameters, and , which we call the class of accordion graphs
, and show that the quartic graphs having the PH-property mentioned by
Alahmadi et al. are in fact members of this general class of accordion graphs.
We also study the PH-property of this class of accordion graphs, at times
considering the pairings of which are also perfect matchings of .
Furthermore, there is a close relationship between accordion graphs and the
Cartesian product of two cycles. Motivated by a recent work by Bogdanowicz
(2015), we give a complete characterisation of those accordion graphs that are
circulant graphs. In fact, we show that is not circulant if and only
if both and are even, such that .Comment: 17 pages, 9 figure
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
Extending a perfect matching to a Hamiltonian cycle
Graph TheoryRuskey and Savage conjectured that in the d-dimensional hypercube, every matching M can be extended to a Hamiltonian cycle. Fink verified this for every perfect matching M, remarkably even if M contains external edges. We prove that this property also holds for sparse spanning regular subgraphs of the cubes: for every d ≥7 and every k, where 7 ≤k ≤d, the d-dimensional hypercube contains a k-regular spanning subgraph such that every perfect matching (possibly with external edges) can be extended to a Hamiltonian cycle. We do not know if this result can be extended to k=4,5,6. It cannot be extended to k=3. Indeed, there are only three 3-regular graphs such that every perfect matching (possibly with external edges) can be extended to a Hamiltonian cycle, namely the complete graph on 4 vertices, the complete bipartite 3-regular graph on 6 vertices and the 3-cube on 8 vertices. Also, we do not know if there are graphs of girth at least 5 with this matching-extendability property