5,692 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
Graphical condensation of plane graphs: a combinatorial approach
The method of graphical vertex-condensation for enumerating perfect matchings
of plane bipartite graph was found by Propp (Theoret. Comput. Sci. 303(2003),
267-301), and was generalized by Kuo (Theoret. Comput. Sci. 319 (2004), 29-57)
and Yan and Zhang (J. Combin. Theory Ser. A, 110(2005), 113-125). In this
paper, by a purely combinatorial method some explicit identities on graphical
vertex-condensation for enumerating perfect matchings of plane graphs (which do
not need to be bipartite) are obtained. As applications of our results, some
results on graphical edge-condensation for enumerating perfect matchings are
proved, and we count the sum of weights of perfect matchings of weighted Aztec
diamond.Comment: 13 pages, 5 figures. accepted by Theoretial Computer Scienc
On the expected number of perfect matchings in cubic planar graphs
A well-known conjecture by Lov\'asz and Plummer from the 1970s asserted that
a bridgeless cubic graph has exponentially many perfect matchings. It was
solved in the affirmative by Esperet et al. (Adv. Math. 2011). On the other
hand, Chudnovsky and Seymour (Combinatorica 2012) proved the conjecture in the
special case of cubic planar graphs. In our work we consider random bridgeless
cubic planar graphs with the uniform distribution on graphs with vertices.
Under this model we show that the expected number of perfect matchings in
labeled bridgeless cubic planar graphs is asymptotically , where
and is an explicit algebraic number. We also
compute the expected number of perfect matchings in (non necessarily
bridgeless) cubic planar graphs and provide lower bounds for unlabeled graphs.
Our starting point is a correspondence between counting perfect matchings in
rooted cubic planar maps and the partition function of the Ising model in
rooted triangulations.Comment: 19 pages, 4 figure
Shortest Reconfiguration of Perfect Matchings via Alternating Cycles
Motivated by adjacency in perfect matching polytopes, we study the shortest reconfiguration problem of perfect matchings via alternating cycles. Namely, we want to find a shortest sequence of perfect matchings which transforms one given perfect matching to another given perfect matching such that the symmetric difference of each pair of consecutive perfect matchings is a single cycle. The problem is equivalent to the combinatorial shortest path problem in perfect matching polytopes. We prove that the problem is NP-hard even when a given graph is planar or bipartite, but it can be solved in polynomial time when the graph is outerplanar
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