480 research outputs found
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
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. ([13]). On the other hand, Chudnovsky and Seymour ([8]) 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 n vertices. Under this model we show that the expected number of perfect matchings in labeled bridgeless cubic planar graphs is asymptotically cγn, where c > 0 and γ ∼ 1.14196 is an explicit algebraic number. We also compute the expected number of perfect matchings in (not 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
Enumeration of maximum matchings of graphs
Counting maximum matchings in a graph is of great interest in statistical
mechanics,
solid-state chemistry, theoretical computer science, mathematics, among other
disciplines. However, it is a challengeable problem to explicitly determine the
number of maximum matchings of general graphs. In this paper, using
Gallai-Edmonds structure theorem, we derive a computing formula for the number
of maximum matching in a graph. According to the formula, we obtain an
algorithm to enumerate maximum matchings of a graph. In particular, The formula
implies that computing the number of maximum matchings of a graph is converted
to compute the number of perfect matchings of some induced subgraphs of the
graph. As an application, we calculate the number of maximum matchings of opt
trees. The result extends a conclusion obtained by Heuberger and Wagner[C.
Heuberger, S. Wagner, The number of maximum matchings in a tree, Discrete Math.
311 (2011) 2512--2542]
Trees maximizing the number of almost-perfect matchings
We characterize the extremal trees that maximize the number of almost-perfect
matchings, which are matchings covering all but one or two vertices, and those
that maximize the number of strong almost-perfect matchings, which are
matchings missing only one or two leaves. We also determine the trees that
minimize the number of maximal matchings. We apply these results to extremal
problems on the weighted Hosoya index for several choices of
vertex-degree-based weight function.Comment: 21 pages, 8 figure
The matching relaxation for a class of generalized set partitioning problems
This paper introduces a discrete relaxation for the class of combinatorial
optimization problems which can be described by a set partitioning formulation
under packing constraints. We present two combinatorial relaxations based on
computing maximum weighted matchings in suitable graphs. Besides providing dual
bounds, the relaxations are also used on a variable reduction technique and a
matheuristic. We show how that general method can be tailored to sample
applications, and also perform a successful computational evaluation with
benchmark instances of a problem in maritime logistics.Comment: 33 pages. A preliminary (4-page) version of this paper was presented
at CTW 2016 (Cologne-Twente Workshop on Graphs and Combinatorial
Optimization), with proceedings on Electronic Notes in Discrete Mathematic
Random multi-index matching problems
The multi-index matching problem (MIMP) generalizes the well known matching
problem by going from pairs to d-uplets. We use the cavity method from
statistical physics to analyze its properties when the costs of the d-uplets
are random. At low temperatures we find for d>2 a frozen glassy phase with
vanishing entropy. We also investigate some properties of small samples by
enumerating the lowest cost matchings to compare with our theoretical
predictions.Comment: 22 pages, 16 figure
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