17 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
Enumeration of labelled 4-regular planar graphs
We present the first combinatorial scheme for counting labelled 4-regular
planar graphs through a complete recursive decomposition. More precisely, we
show that the exponential generating function of labelled 4-regular planar
graphs can be computed effectively as the solution of a system of equations,
from which the coefficients can be extracted. As a byproduct, we also enumerate
labelled 3-connected 4-regular planar graphs, and simple 4-regular rooted maps
Asymptotic study of regular planar graphs
The central topic of this dissertation is the study of some families of regular planar graphs and maps.
We are in particular interested in their asymptotic enumeration in order to understand of the associated uniform random model.
In a first part, we give both an exact and an asymptotic enumeration of labelled cubic planar graphs, multigraphs and simple maps, via a recursive scheme following the iterative decompositon of a graph in smaller components of higher connecttivity.
In the second part, we apply those results to the study a the uniform random labelled cubic planar graph.
We compute for instance the probability of connectivity, and prove that some significant parameters are distributed following a Gaussian limit law: the numbers of cut-vertices, isthmuses, blocks, cherries, near-bricks, and triangles.
In the third and last part, we develop the first recursive combinatorial scheme to enumerate 4-regular labelled planar graphs.
This scheme is based on a decomposition in terms of connectivity, similar to that of cubic planar graphs, which leads to the exact enumeration of 4-regular planar graphs and simple maps.Das zentrale Thema dieser Dissertation sind Familien von regulÀren planaren Graphen und Karten.
Insbesondere sind wir an daran interessiert, diese zu zĂ€hlen und die ZusammenhĂ€nge zu deren zufĂ€lligen GegenstĂŒcken zu erforschen.
Im ersten Teil geben wir sowohl eine rekursive als auch eine asymptotische AbzÀhlung von kubischen, planaren Graphen, Multigraphen und einfachen Karten, durch eine Dekomposition entlang deren Komponenten.
Im zweiten Teil wenden wir diese Resultate auf zufÀllige kubische planare Graphen an.
Insbesondere berechnen wir die Wahrscheinlichkeit von ZusammenhÀngigkeit, und beweisen das einige bedeutende Parameter normalverteilt sind: die Anzahl der cut-vertices, isthmuses, Blöcke, cherries, near-bricks und Dreiecke.
Im dritten und letzten Teil entwickeln wir das erste kombinatorisches Schema, basierend auf einem Dekompositionsschema das Ă€hnlich zu dem im Kontext von kubischen planaren Graphen ist, das zur rekursiven AbzĂ€hlung von 4-regulĂ€ren planaren Graphen und einfachen Karten fĂŒhrt
Variants of Plane Diameter Completion
The {\sc Plane Diameter Completion} problem asks, given a plane graph and
a positive integer , if it is a spanning subgraph of a plane graph that
has diameter at most . We examine two variants of this problem where the
input comes with another parameter . In the first variant, called BPDC,
upper bounds the total number of edges to be added and in the second, called
BFPDC, upper bounds the number of additional edges per face. We prove that
both problems are {\sf NP}-complete, the first even for 3-connected graphs of
face-degree at most 4 and the second even when on 3-connected graphs of
face-degree at most 5. In this paper we give parameterized algorithms for both
problems that run in steps.Comment: Accepted in IPEC 201
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 cÎłn, where c > 0 and Îł ⌠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
Asymptotic enumeration of labelled 4-regular planar graphs
Building on previous work by the present authors [Proc. London Math. Soc.
119(2):358--378, 2019], we obtain a precise asymptotic estimate for the number
of labelled 4-regular planar graphs. Our estimate is of the form , where is a constant and is the radius of convergence of the generating function , and conforms to the universal pattern obtained previously in the
enumeration of planar graphs. In addition to analytic methods, our solution
needs intensive use of computer algebra in order to work with large systems of
polynomials equations. In particular, we use evaluation and Lagrange
interpolation in order to compute resultants of multivariate polynomials. We
also obtain asymptotic estimates for the number of 2- and 3-connected 4-regular
planar graphs, and for the number of 4-regular simple maps, both connected and
2-connected.Comment: 23 pages, including 5 pages of appendix. Corrected titl
Further results on random cubic planar graphs
We provide precise asymptotic estimates for the number of several classes of labeled cubic planar graphs, and we analyze properties of such random graphs under the uniform distribution. This model was first analyzed by Bodirsky and coworkers. We revisit their work and obtain new results on the enumeration of cubic planar graphs and on random cubic planar graphs. In particular, we determine the exact probability of a random cubic planar graph being connected, and we show that the distribution of the number of triangles in random cubic planar graphs is asymptotically normal with linear expectation and variance. To the best of our knowledge, this is the first time one is able to determine the asymptotic distribution for the number of copies of a fixed graph containing a cycle in classes of random planar graphs arising from planar maps.Peer ReviewedPostprint (author's final draft
Maximal independent sets and maximal matchings in series-parallel and related graph classes
We provide combinatorial decompositions as well as asymptotic tight estimates for two maximal parameters: the number and average size of maximal independent sets and maximal matchings in seriesparallel graphs (and related graph classes) with n vertices. In particular, our results extend previous results of Meir and Moon for trees [Meir, Moon: On maximal independent sets of nodes in trees, Journal of Graph Theory 1988]. We also show that these two parameters converge to a central limit law.Postprint (author's final draft