77 research outputs found
Matching Preclusion and Conditional Matching Preclusion Problems for Twisted Cubes
The matching preclusion number of a graph is the minimum
number of edges whose deletion results in a graph that has neither
perfect matchings nor almost-perfect matchings. For many interconnection
networks, the optimal sets are precisely those induced by a
single vertex. Recently, the conditional matching preclusion number
of a graph was introduced to look for obstruction sets beyond those
induced by a single vertex. It is defined to be the minimum number
of edges whose deletion results in a graph with no isolated vertices
that has neither perfect matchings nor almost-perfect matchings. In
this paper, we find the matching preclusion number and the conditional matching preclusion number for twisted cubes, an improved
version of the well-known hypercube. Moreover, we also classify all
the optimal matching preclusion sets
On the central levels problem
The \emph{central levels problem} asserts that the subgraph of the -dimensional hypercube induced by all bitstrings with at least many 1s and at most many 1s, i.e., the vertices in the middle levels, has a Hamilton cycle for any and .
This problem was raised independently by Buck and Wiedemann, Savage, Gregor and {\v{S}}krekovski, and by Shen and Williams, and it is a common generalization of the well-known \emph{middle levels problem}, namely the case , and classical binary Gray codes, namely the case .
In this paper we present a general constructive solution of the central levels problem.
Our results also imply the existence of optimal cycles through any sequence of consecutive levels in the -dimensional hypercube for any and .
Moreover, extending an earlier construction by Streib and Trotter, we construct a Hamilton cycle through the -dimensional hypercube, , that contains the symmetric chain decomposition constructed by Greene and Kleitman in the 1970s, and we provide a loopless algorithm for computing the corresponding Gray code
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
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