Related papers: Link colorings and the Goeritz matrix
This paper provides a relationship between a geometric structure of a suspended tree and the number of link components of the associated link diagram.
We introduce a new combinatorial object called a web world that consists of a set of web diagrams. The diagrams of a web world are generalizations of graphs, and each is built on the same underlying graph. Instead of ordinary vertices the…
This chapter is an introduction to the connection between random matrices and maps, i.e graphs drawn on surfaces. We concentrate on the one-matrix model and explain how it encodes and allows to solve a map enumeration problem.
In this paper, we show that a link which has a positive and almost alternating diagram is alternating, besides that a positive and non-alternating Montesinos link has an almost positive-alternating diagram.
We introduce multi-tribrackets, algebraic structures for region coloring of diagrams of knots and links with different operations at different kinds of crossings. In particular we consider the case of component multi-tribrackets which have…
A proper $q$-coloring of a graph is an assignment of one of $q$ colors to each vertex of the graph so that adjacent vertices are colored differently. Sample uniformly among all proper $q$-colorings of a large discrete cube in the integer…
We introduce an up-down coloring of a virtual-link diagram. The colorabilities give a lower bound of the minimum number of Reidemeister moves of type II which are needed between two 2-component virtual-link diagrams. By using the notion of…
Scheduling wireless links for simultaneous activation in such a way that all transmissions are successfully decoded at the receivers and moreover network capacity is maximized is a computationally hard problem. Usually it is tackled by…
An edge-locating coloring of a simple connected graph $G$ is a partition of its edge set into matchings such that the vertices of $G$ are distinguished by the distance to the matchings. The minimum number of the matchings of $G$ that admits…
This chapter presents an introduction to graph colouring algorithms. The focus is on vertex-colouring algorithms that work for general classes of graphs with worst-case performance guarantees in a sequential model of computation. The…
We give a simple example showing that a knot or link diagram that lies in the ${\mathbb{Z}}^2$ lattice is not necessarily the projection of a lattice stick knot or link in the ${\mathbb{Z}}^3$ lattice, and we give a necessary and sufficient…
We propose a new approach for defining and searching clusters in graphs that represent real technological or transaction networks. In contrast to the standard way of finding dense parts of a graph, we concentrate on the structure of edges…
For a positive integer k and a graph G, we consider proper vertex-colourings of G with k colours in which all k colours are actually used. We call such a colouring a strong k-colouring. The strong k-colour graph of G, S_k(G), is the graph…
Motivated by the Erd\H{o}s-Faber-Lov\'asz (EFL) conjecture for hypergraphs, we consider the list edge coloring of linear hypergraphs. We discuss several conjectures for list edge coloring linear hypergraphs that generalize both EFL and…
Motivated by the Erdos-Faber Lovasz conjecture (EFL) for hypergraphs, we explore relationships between several conjectures on the edge coloring of linear hypergraphs. In particular, we are able to increase the class of hypergraphs for which…
Proper vertex colorings of a graph are related to its boundary map, also called its signed vertex-edge incidence matrix. The vertex Laplacian of a graph, a natural extension of the boundary map, leads us to introduce nowhere-harmonic…
A link diagram is said to be lune-free if, when viewed as a 4-regular plane graph it does not have multiple edges between any pair of nodes. We prove that any colored link diagram is equivalent to a colored lune-free diagram with the same…
Coloured graphical models are Gaussian statistical models determined by an undirected coloured graph. These models can be described by linear spaces of symmetric matrices. We outline a relationship between the symmetries of the graph and…
We show that any two diagrams of the same knot or link are connected by a sequence of Reidemeister moves which are sorted by type.
We present old and recent results on rank problems and linearizability of geodesic planar webs.