5,649 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
Efficient Generation of Stable Planar Cages for Chemistry
In this paper we describe an algorithm which generates all colored planar
maps with a good minimum sparsity from simple motifs and rules to connect them.
An implementation of this algorithm is available and is used by chemists who
want to quickly generate all sound molecules they can obtain by mixing some
basic components.Comment: 17 pages, 7 figures. Accepted at the 14th International Symposium on
Experimental Algorithms (SEA 2015
A Victorian Age Proof of the Four Color Theorem
In this paper we have investigated some old issues concerning four color map
problem. We have given a general method for constructing counter-examples to
Kempe's proof of the four color theorem and then show that all counterexamples
can be rule out by re-constructing special 2-colored two paths decomposition in
the form of a double-spiral chain of the maximal planar graph. In the second
part of the paper we have given an algorithmic proof of the four color theorem
which is based only on the coloring faces (regions) of a cubic planar maps. Our
algorithmic proof has been given in three steps. The first two steps are the
maximal mono-chromatic and then maximal dichromatic coloring of the faces in
such a way that the resulting uncolored (white) regions of the incomplete
two-colored map induce no odd-cycles so that in the (final) third step four
coloring of the map has been obtained almost trivially.Comment: 27 pages, 18 figures, revised versio
Thoughts on Barnette's Conjecture
We prove a new sufficient condition for a cubic 3-connected planar graph to
be Hamiltonian. This condition is most easily described as a property of the
dual graph. Let be a planar triangulation. Then the dual is a cubic
3-connected planar graph, and is bipartite if and only if is
Eulerian. We prove that if the vertices of are (improperly) coloured blue
and red, such that the blue vertices cover the faces of , there is no blue
cycle, and every red cycle contains a vertex of degree at most 4, then is
Hamiltonian.
This result implies the following special case of Barnette's Conjecture: if
is an Eulerian planar triangulation, whose vertices are properly coloured
blue, red and green, such that every red-green cycle contains a vertex of
degree 4, then is Hamiltonian. Our final result highlights the
limitations of using a proper colouring of as a starting point for proving
Barnette's Conjecture. We also explain related results on Barnette's Conjecture
that were obtained by Kelmans and for which detailed self-contained proofs have
not been published.Comment: 12 pages, 7 figure
Fracton Topological Order, Generalized Lattice Gauge Theory and Duality
We introduce a generalization of conventional lattice gauge theory to
describe fracton topological phases, which are characterized by immobile,
point-like topological excitations, and sub-extensive topological degeneracy.
We demonstrate a duality between fracton topological order and interacting spin
systems with symmetries along extensive, lower-dimensional subsystems, which
may be used to systematically search for and characterize fracton topological
phases. Commutative algebra and elementary algebraic geometry provide an
effective mathematical toolset for our results. Our work paves the way for
identifying possible material realizations of fracton topological phases.Comment: 9 pages, 4 figures; 8 pages of appendices, 3 figure
An exact solution on the ferromagnetic Face-Cubic spin model on a Bethe lattice
The lattice spin model with --component discrete spin variables restricted
to have orientations orthogonal to the faces of -dimensional hypercube is
considered on the Bethe lattice, the recursive graph which contains no cycles.
The partition function of the model with dipole--dipole and
quadrupole--quadrupole interaction for arbitrary planar graph is presented in
terms of double graph expansions. The latter is calculated exactly in case of
trees. The system of two recurrent relations which allows to calculate all
thermodynamic characteristics of the model is obtained. The correspondence
between thermodynamic phases and different types of fixed points of the RR is
established. Using the technique of simple iterations the plots of the zero
field magnetization and quadrupolar moment are obtained. Analyzing the regions
of stability of different types of fixed points of the system of recurrent
relations the phase diagrams of the model are plotted. For the phase
diagram of the model is found to have three tricritical points, whereas for there are one triple and one tricritical points.Comment: 20 pages, 7 figure
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