Related papers: Enumeration of 4-regular one-face maps

We introduce S-nondegenerate formal CR maps and establish their convergence (revision of a preliminary version).

We give a rational form of a generic two-dimensional "quad" map, containing the so-called $Q_4$ case, but whose coefficients are free. Its integrability is proved using the calculation of algebraic entropy.

We describe constructions of infinite graphs which are not representable as integral graphs in the plane, addressing a question of Erd\H{o}s. We also mention some related problems.

Given a finite or infinite planar graph all of whose faces have degree 4, we study embeddings in the plane in which all edges have length 1, that is, in which every face is a rhombus. We give a necessary and sufficient condition for the…

We prove that the facial nonrepetitive chromatic number of any outerplanar graph is at most 11 and of any planar graph is at most 22.

We give a practical formula for counting irreducible nodal genus-three plane curves that a fixed generic complex structure on the normalization. As an intermediate step, we enumerate rational plane curves that have a $(3,4)$-cusp.

A connected 3-valent plane graph, whose faces are $q$- or 6-gons only, is called a {\em graph $q_n$}. We classify all graphs $4_n$, which are isometric subgraphs of a $m$-hypercube $H_m$.

Considering regions in a map to be adjacent when they have nonempty intersection (as opposed to the traditional view requiring intersection in a linear segment) leads to the concept of a facially complete graph: a plane graph that becomes…

We present an enumeration of orientably-regular maps with automorphism group isomorphic to the twisted linear fractional group $M(q^2)$ for any odd prime power $q$.

We show an asymptotic estimate for the number of labelled planar graphs on $n$ vertices. We also find limit laws for the number of edges, the number of connected components, and other parameters in random planar graphs.

We introduce the concept of alternate-edge-colourings for maps, and study highly symmetric examples of such maps. Edge-biregular maps of type $(k,l)$ occur as smooth normal quotients of a particular index two subgroup of $T_{k,l}$, the full…

We produce curves with a record number of points over the finite fields with $4$, $9$, $16$ and $25$ elements, as unramified abelian covers of algebraic curves.

We give a full characterisation of the symmetries of unlabelled triangulations and derive a constructive decomposition of unlabelled triangulations depending on their symmetries. As an application of these results we can deduce a complete…

We present a natural extension of the notion of nondegenerate rational maps (quadrirational maps) to arbitrary dimensions. We refer to these maps as $2^n-$rational maps. In this note we construct a rich family of $2^n-$rational maps. These…

This paper initiates the explicit study of face numbers of matroid polytopes and their computation. We prove that, for the large class of split matroid polytopes, their face numbers depend solely on the number of cyclic flats of each rank…

We compute the topological mapping class group of every compact, simply connected, topological 4-manifold. This was previously only known in the closed case. If the 4-manifold is smooth, we deduce an analogous description of the stable…

We describe computational results about undirected graphs having $10$ vertices and automorphism group isomorphic to $\mathbb{Z}/4\mathbb{Z}$.

Let $G$ be a quadrangulation on an orientable surface and let $g$ be a proper vertex-$4$-coloring of $G$. A face $F$ of $G$ is said to be a rainbow-face if all four distinct colors appear on its boundary. A $(c_1,c_2,c_3,c_4)$-face in $G$…

We compute the oriented cobordism group of fold maps of 4-manifolds into the space with all the possible restrictions (and also with no restriction) to the singular fibers. We also give geometric invariants which describe completely the…

Uniform Lie algebras are combinatorially defined two-step nilpotent Lie algebras which can be used to define Einstein solvmanifolds. These Einstein spaces often have nontrivial isotropy groups. We derive basic properties of uniform Lie…