Related papers: Generalized Heawood Graphs and Triangulations of T…
Honeycomb toroidal graphs are trivalent Cayley graphs on generalized dihedral groups. We examine the two historical threads leading to these graphs, some of the properties that have been established, and some open problems.
We prove a variation on Heawood list-coloring for graphs on surfaces, modeled on Thomassen's planar 5-list-coloring theorem. For epsilon>0 define the Heawood number to be H(epsilon)=Floor((7+Sqrt[24*epsilon+1])/2). We prove that, except for…
Tensor models are used nowadays for implementing a fundamental theory of quantum gravity. We define here a polynomial $\mathcal T$ encoding the supplementary topological information. This polynomial is a natural generalization of the…
Regular colored graphs are dual representations of pure colored D-dimensional complexes. These graphs can be classified with respect to an integer, their degree, much like maps are characterized by the genus. We analyse the structure of…
We study finite graphs embedded in oriented surfaces by associating a polynomial to it. The tools used in developing a theory of such graph polynomials are algebraic topological while the polynomial itself is inspired from ideas arising in…
In this paper we introduce a class of hypergraphs that we call chordal. We also extend the definition of triangulated hypergraphs, given in \cite{VT}, so that a triangulated hypergraph, according to our definition, is a natural…
A categorical formalism for directed graphs is introduced, featuring natural notions of morphisms and subgraphs, and leading to two elementary descriptions of the free-properad monad, first in terms of presheaves on elementary graphs,…
In this paper we consider the following three coloring concepts for digraphs. First of all, the generalized coloring concept, in which the same colored vertices of a digraph induce a subdigraph that satisfies a given digraph property.…
This report is an introduction to mathematical map colouring and the problems posed by Heawood in his paper of 1890. There will be a brief discussion of the Map Colour Theorem; then we will move towards investigating empire maps in the…
This paper introduces a geometric representation of hypergraphs by representing hyperedges as simplices. Building on this framework, we employ homotopy groups to analyze the topological structure of hypergraphs embedded in high-dimensional…
The \emph{coloring number} $\mathrm{col}(G)$ of a graph $G$, which is equal to the \emph{degeneracy} of $G$ plus one, provides a very useful measure for the uniform sparsity of $G$. The coloring number is generalized by three series of…
A Schnyder wood is an orientation and coloring of the edges of a planar map satisfying a simple local property. We propose a generalization of Schnyder woods to graphs embedded on the torus with application to graph drawing. We prove…
This paper proposes a family of permutation-invariant graph embeddings, generalizing the Skew Spectrum of graphs of Kondor & Borgwardt (2008). Grounded in group theory and harmonic analysis, our method introduces a new class of graph…
A tetravalent $2$-arc-transitive graph of order $728$ is either the known $7$-arc-transitive incidence graph of the classical generalized hexagon $GH(3,3)$ or a normal cover of a $2$-transitive graph of order $182$ denoted $A[182,1]$ or…
A graph with a semiregular group of automorphisms can be thought of as the derived cover arising from a voltage graph. Since its inception, the theory of voltage graphs and their derived covers has been a powerful tool used in the study of…
Graph colouring is a combinatorial optimisation problem with applications in several important domains, including sports scheduling, cartography, street map navigation, and timetabling. It is also of significant theoretical interest and a…
We demonstrate how to generalize two of the most well-known random graph models, the classic random graph, and random graphs with a given degree distribution, by the introduction of hidden variables in the form of extra degrees of freedom,…
In this paper, we use the theory of Riordan matrices to introduce the notion of an oriented Riordan graph. The oriented Riordan graphs are a far-reaching generalization of the well known and well studied Toeplitz oriented graphs and…
In this paper we resolve the complexity of the isomorphism problem on all but finitely many of the graph classes characterized by two forbidden induced subgraphs. To this end we develop new techniques applicable for the structural and…
{\em Honeycomb toroidal graphs} are a family of cubic graphs determined by a set of three parameters, that have been studied over the last three decades both by mathematicians and computer scientists. They can all be embedded on a torus and…