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 $n$ vertices. Under this model we show that the expected number of perfect matchings in labeled bridgeless cubic planar graphs is asymptotically $c\gamma^n$, where $c>0$ and $\gamma \sim 1.14196$ 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

{\Gamma}-species, quotients, and graph enumeration

The theory of {\Gamma}-species is developed to allow species-theoretic study of quotient structures in a categorically rigorous fashion. This new approach is then applied to two graph-enumeration problems which were previously unsolved in the unlabeled case-bipartite blocks and general k-trees.Comment: 84 pages, 10 figures, dissertatio

Combinatorial species and graph enumeration

In enumerative combinatorics, it is often a goal to enumerate both labeled and unlabeled structures of a given type. The theory of combinatorial species is a novel toolset which provides a rigorous foundation for dealing with the distinction between labeled and unlabeled structures. The cycle index series of a species encodes the labeled and unlabeled enumerative data of that species. Moreover, by using species operations, we are able to solve for the cycle index series of one species in terms of other, known cycle indices of other species. Section 3 is an exposition of species theory and Section 4 is an enumeration of point-determining bipartite graphs using this toolset. In Section 5, we extend a result about point-determining graphs to a similar result for point-determining {\Phi}-graphs, where {\Phi} is a class of graphs with certain properties. Finally, Appendix A is an expository on species computation using the software Sage [9] and Appendix B uses Sage to calculate the cycle index series of point-determining bipartite graphs.Comment: 39 pages, 16 figures, senior comprehensive project at Carleton Colleg

Planar maps as labeled mobiles

We extend Schaeffer's bijection between rooted quadrangulations and well-labeled trees to the general case of Eulerian planar maps with prescribed face valences, to obtain a bijection with a new class of labeled trees, which we call mobiles. Our bijection covers all the classes of maps previously enumerated by either the two-matrix model used by physicists or by the bijection with blossom trees used by combinatorists. Our bijection reduces the enumeration of maps to that, much simpler, of mobiles and moreover keeps track of the geodesic distance within the initial maps via the mobiles' labels. Generating functions for mobiles are shown to obey systems of algebraic recursion relations.Comment: 31 pages, 17 figures, tex, lanlmac, epsf; improved tex

Determinant Sums for Undirected Hamiltonicity

We present a Monte Carlo algorithm for Hamiltonicity detection in an $n$-vertex undirected graph running in $O^*(1.657^{n})$ time. To the best of our knowledge, this is the first superpolynomial improvement on the worst case runtime for the problem since the $O^*(2^n)$ bound established for TSP almost fifty years ago (Bellman 1962, Held and Karp 1962). It answers in part the first open problem in Woeginger's 2003 survey on exact algorithms for NP-hard problems. For bipartite graphs, we improve the bound to $O^*(1.414^{n})$ time. Both the bipartite and the general algorithm can be implemented to use space polynomial in $n$. We combine several recently resurrected ideas to get the results. Our main technical contribution is a new reduction inspired by the algebraic sieving method for $k$-Path (Koutis ICALP 2008, Williams IPL 2009). We introduce the Labeled Cycle Cover Sum in which we are set to count weighted arc labeled cycle covers over a finite field of characteristic two. We reduce Hamiltonicity to Labeled Cycle Cover Sum and apply the determinant summation technique for Exact Set Covers (Bj\"orklund STACS 2010) to evaluate it.Comment: To appear at IEEE FOCS 201

{\Gamma}-species and the enumeration of k-trees

We study the class of graphs known as k-trees through the lens of Joyal's theory of combinatorial species (and an equivariant extension known as '$\Gamma$-species' which incorporates data about 'structural' group actions). This culminates in a system of recursive functional equations giving the generating function for unlabeled k-trees which allows for fast, efficient computation of their numbers. Enumerations up to k = 10 and n = 30 (for a k-tree with (n+k-1) vertices) are included in tables, and Sage code for the general computation is included in an appendix.Comment: 26 pages; includes Python cod