We design an O(m) algorithm to find a minimum weighted colouring and a maximum weighted clique of a perfectly ordered graph. We also present two O(n’) algorithms to find a minimum weighted colouring of a comparability graph and of a triangulated graph. Our colouring algorithms use an algorithm to find a stable set meeting all maximal (with respect to set inclusion) cliques of a perfectly ordere...

A graph is (H1,H2)-free for a pair of graphsH1, H2 if it contains no induced subgraph isomorphic toH1 orH2. In 2001, Král’, Kratochvíl, Tuza, and Woeginger initiated a study into the complexity of Colouring for (H1,H2)free graphs. Since then, others have tried to complete their study, but many cases remain open. We focus on those (H1,H2)-free graphs where H2 is H1, the complement of H1. As thes...

A graph is (H1,H2)-free for a pair of graphsH1, H2 if it contains no induced subgraph isomorphic toH1 orH2. In 2001, Král’, Kratochvíl, Tuza, and Woeginger initiated a study into the complexity of Colouring for (H1,H2)free graphs. Since then, others have tried to complete their study, but many cases remain open. We focus on those (H1,H2)-free graphs where H2 is H1, the complement of H1. As thes...

We define k-diverse colouring of a graph to be a proper vertex colouring in which every vertex x, sees min{k, d(x)} different colours in its neighbors. We show that for given k there is an f(k) for which every planar graph admits a k-diverse colouring using at most f(k) colours. Then using this colouring we obtain a K5-free graph H for which every planar graph admits a homomorphism to it, thus ...

Perfect colouring of isonemal fabrics by thin and thick striping of warp and weft with more than two colours is introduced. Conditions that prevent perfect colouring by striping are derived, and it is shown that avoiding the preventing conditions is sufficient to allow perfect colouring. Examples of thick striping in all possible species are

The Minimum Sum Colouring Problem is an NP-hard problem derived from the well-known graph colouring problem. It consists in finding a proper colouring which minimizes the sum of the assigned colours rather than the number of those colours. This problem often arises in scheduling and resource allocation. Mainly incomplete approaches were proposed, but Integer Linear Programming (ILP) and Constra...

Bagging of female inflorescence following pollination has recently received growers attention. Although this process occurs early in the season but affects fruit quality, yield, and marketability. Date palm growers have used to use grocery paper bags for this bagging process. Since inflorescence bagging causes what is called the greenhouse effect, it was very important to investigate the effect...

A list-colouring of a graph is an assignment of a colour to each vertex v from its own list L(v) of colours. Instead of colouring vertices we may want to colour other elements of a graph such as edges, faces, or any combination of vertices, edges and faces. In this thesis we will study several of these different types of list-colouring, each for the class of a near-outerplanar graphs. Since a g...

We introduce a novel representation for the graph colouring problem, called the Integer Merge Model, which aims to reduce the time complexity of graph colouring algorithms. Moreover, this model provides useful information to aid in the creation of heuristics that can make the colouring process even faster. It also serves as a compact definition for the description of graph colouring algorithms....

In this paper, we consider a new edge colouring problem motivated by wireless mesh networks optimization: the proportional edge colouring problem. Given a graph G with positive weights associated to its edges, we want to find a proper edge colouring which assigns to each edge at least a proportion (given by its weight) of all the colours. If such colouring exists, we want to find one using the ...