Related papers: Coloured Hard-Dimers
In this note we discuss the relationship between the generating functions of some Hermite polynomials $H$, $ \sum\limits_{j=0}^\infty H_{j\cdot n}(u) z^n/n!$, generalized Airy-Heat equations…
This document presents a combinatorial framework for analyzing assembly systems using generating functions. We explore the theory through concrete examples, such as linear polymers, and develop recursive equations to characterize valid…
The Art Gallery Problem is one of the most well-known problems in Computational Geometry, with a rich history in the study of algorithms, complexity, and variants. Recently there has been a surge in experimental work on the problem. In this…
A computer program for evaluating colour factors of QCD Feynman diagrams is presented, and illustrative examples on how to use the program to calculate non trivial colour factors are given. The program and the discussion in this paper is…
Graph drawing research traditionally focuses on producing geometric embeddings of graphs satisfying various aesthetic constraints. After the geometric embedding is specified, there is an additional step that is often overlooked or ignored:…
The chromatic number of signed graphs is defined recently. The coloring and clique problem of interval graphs has been studied and polynomial time algorithms are established. Here we consider these problems for signed interval graphs and…
We show that the half-perimeter generating functions for the number of Wicketed and Gated Ferrers diagrams is algebraic. Furthermore, the generating function of the Wicketed Ferrers diagrams is closely related to the generating function of…
We give a generating function for the number of graphs with given numerical properties and prescribed weighted number of connected components. As an application, we give a generating function for the number of bipartite graphs of given…
Recent experimental results on inclusive diffractive scattering and on exclusive vector meson production are reviewed. The dynamical picture of hard diffraction emerging in perturbative QCD is highlighted.
For any fixed surface Sigma of genus g, we give an algorithm to decide whether a graph G of girth at least five embedded in Sigma is colorable from an assignment of lists of size three in time O(|V(G)|). Furthermore, we can allow a subgraph…
We show a method how to convert any graph into the binary number and vice versa. We derive upper bound for maximum number of graphs, that, have fixed number of vertices and can be colored with n colors (n is any given number). Proof for the…
In this paper, we study the achromatic and the pseudoachromatic numbers of planar and outerplanar graphs as well as planar graphs of girth 4 and graphs embedded on a surface. We give asymptotically tight results and lower bounds for maximal…
Several interesting generative learning algorithms involve a complex probability distribution over many random variables, involving intractable normalization constants or latent variable normalization. Some of them may even not have an…
Two colourings of a graph are orthogonal if they have the property that when two vertices are coloured with the same colour in one colouring, then those vertices receive distinct colours in the other colouring. In this paper, orthogonal…
We construct for every integer $k\geq 3$ and every real $\mu\in(0, \frac{k-1}{k})$ a set of integers $X=X(k, \mu)$ which, when coloured with finitely many colours, contains a monochromatic $k$-term arithmetic progression, whilst every…
The problem of vertex coloring in random graphs is studied using methods of statistical physics and probability. Our analytical results are compared to those obtained by exact enumeration and Monte-Carlo simulations. We critically discuss…
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…
The dimer tiling problem asks in how many ways can the edges of a graph be covered by dimers so that each site is covered once. In the special case of a planar graph, this problem has a solution in terms of a free fermionic field theory. We…
Graph colorings are becoming an increasingly useful family of mathematical models for a broad range of applications, such as time tabling and scheduling, frequency assignment, register allocation, computer security and so on. Graph proper…
The purpose of this note is to draw attention to problems related to a concept called majority colouring recently studied by Kreutzer, Oum, Seymour, van der Zypen and Wood. They raised a problem of determining, for a natural number $k$, the…