Cuts in matchings of 3-connected cubic graphs

We discuss conjectures on Hamiltonicity in cubic graphs (Tait, Barnette, Tutte), on the dichromatic number of planar oriented graphs (Neumann-Lara), and on even graphs in digraphs whose contraction is strongly connected (Hochst\"attler). We show that all of them fit into the same framework related to cuts in matchings. This allows us to find a counterexample to the conjecture of Hochst\"attler and show that the conjecture of Neumann-Lara holds for all planar graphs on at most 26 vertices. Finally, we state a new conjecture on bipartite cubic oriented graphs, that naturally arises in this setting.Comment: 12 pages, 5 figures, 1 table. Improved expositio

Combinatorics of bicubic maps with hard particles

We present a purely combinatorial solution of the problem of enumerating planar bicubic maps with hard particles. This is done by use of a bijection with a particular class of blossom trees with particles, obtained by an appropriate cutting of the maps. Although these trees have no simple local characterization, we prove that their enumeration may be performed upon introducing a larger class of "admissible" trees with possibly doubly-occupied edges and summing them with appropriate signed weights. The proof relies on an extension of the cutting procedure allowing for the presence on the maps of special non-sectile edges. The admissible trees are characterized by simple local rules, allowing eventually for an exact enumeration of planar bicubic maps with hard particles. We also discuss generalizations for maps with particles subject to more general exclusion rules and show how to re-derive the enumeration of quartic maps with Ising spins in the present framework of admissible trees. We finally comment on a possible interpretation in terms of branching processes.Comment: 41 pages, 19 figures, tex, lanlmac, hyperbasics, epsf. Introduction and discussion/conclusion extended, minor corrections, references adde

A note on Barnette's conjecture

Barnette conjectured that each planar, bipartite, cubic, and 3-connected graph is hamiltonian. We prove that this conjecture is equivalent to the statement that there is a constant c > 0 such that each graph G of this class contains a path on at least c |V(G) | vertices

Legendrian Weaves: N-graph Calculus, Flag Moduli and Applications

We study a class of Legendrian surfaces in contact five-folds by encoding their wavefronts via planar combinatorial structures. We refer to these surfaces as Legendrian weaves, and to the combinatorial objects as N-graphs. First, we develop a diagrammatic calculus which encodes contact geometric operations on Legendrian surfaces as multi-colored planar combinatorics. Second, we present an algebraic-geometric characterization for the moduli space of microlocal constructible sheaves associated to these Legendrian surfaces. Then we use these N-graphs and the flag moduli description of these Legendrian invariants for several new applications to contact and symplectic topology. Applications include showing that any finite group can be realized as a subfactor of a 3-dimensional Lagrangian concordance monoid for a Legendrian surface in the 1-jet space of the two-sphere, a new construction of infinitely many exact Lagrangian fillings for Legendrian links in the standard contact three-sphere, and performing rational point counts over finite fields that distinguish Legendrian surfaces in the standard five-dimensional Darboux chart. In addition, the manuscript develops the notion of Legendrian mutation, studying microlocal monodromies and their transformations. The appendix illustrates the connection between our N-graph calculus for Lagrangian cobordisms and Elias-Khovanov-Williamson's Soergel Calculus.Comment: 114 Pages, 105 Figure