Abstract:
We define a new colouring for a hypergraph, in particular for a graph. Such a method is a partition of the vertex-set of a hypergraph, in particular of a graph. However, it is more intrinsically linked to the geometric structure of the hypergraph and therefore enables us to obtain stronger results than in the classical case. For instance, we prove theorems concerning 3-colourings, 4-colourings and 5-colourings, while we have no analogous results in the classical case. Moreover, we prove that there are no semi-hamiltonian regular simple graphs admitting a hamiltonian 1-colouring. Finally, we characterize the above graphs admitting a hamiltonian 2-colouring and a hamiltonian 3-colouring.

Abstract:
A proper edge colouring f of a graph G is called acyclic if there are no bichromatic cycles in the graph. The acyclic edge chromatic number or acyclic chromatic index, denoted by , is the minimum number of colours in an acyclic edge colouring of G. In this paper, we discuss the acyclic edge colouring of middle, central, total and line graphs of prime related star graph families. Also exact values of acyclic chromatic indices of such graphs are derived and some of their structural properties are discussed.

Abstract:
In this paper, we discuss about the b-colouring and b-chromatic number for middle graph of Cycle, Path, Fan graph and Wheel graph denoted as M[C_{n}],M[P_{n}],M[F_{1,n}] and M[W_{n}] .

Abstract:
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. To verify the potential of the model, we use it in the complete algorithm DSATUR, and in two version of an incomplete approximation algorithm; an evolutionary algorithm and the same evolutionary algorithm extended with guiding heuristics. Both theoretical and empirical results are provided investigation is performed to show an increase in the efficiency of solving graph colouring problems. Two problem suites were used for the empirical evidence: a set of practical problem instances and a set of hard problem instances from the phase transition.

Abstract:
Exam timetable is difficult to be done manually due to several factors such as dual academic calendar, larger student enrolments, constraints among invigilators and limitations of resources. At tertiary education, preparing exam timetable is very critical in order to ensure that all students are able to sit for the exam of every subject that they have registered without any clashing and only sit for one exam at one time. The lecturers who are also the invigilators as well need to be considered as one of the elements in the development of exam timetable as they are required to be in one venue at one time. Therefore, a good time table needs to ensure that the students and invigilators are able to commit their roles accordingly during the exam period. But the main problem is the duration of the exam which will be extended to fulfil all the requirements. This study presents a solution method intended for reducing exam duration in Centre for Foundation Studies and Extension Education (FOSEE), Multimedia University (MMU), Malaysia. The method of solution is using heuristic approaches that include graph colouring, clustering and sequential heuristic. The discussions were focused on constraints among invigilators and the approach is tested on real-world exam timetabling problems.

Abstract:
This paper presents a new algorithm namely Vertex Merge Algorithm(VMA) for channel allocation in Direct Sequence Spread Spectrum(DSSS). VMA try to solve channel allocation based on graph theory. Results from the simulation study reveal that the new graph model can provide reduce the channel required.

Abstract:
Line labelling has been used to determine whether a two-dimensional (2D) line drawing object is a possible or impossible representation of a three-dimensional (3D) solid object. However, the results are not sufficiently robust because the existing line labelling methods do not have any validation method to verify their own result. In this research paper, the concept of graph colouring is applied to a validation technique for a labelled 2D line drawing. As a result, a graph colouring algorithm for validating labelled 2D line drawings is presented. A high-level programming language, MATLAB R2009a, and two primitive 2D line drawing classes, prism and pyramid are used to show how the algorithms can be implemented. The proposed algorithm also shows that the minimum number of colours needed to colour the labelled 2D line drawing object is equal to 3 for prisms and for pyramids, where n is the number of vertices (junctions) in the pyramid objects.

Abstract:
In this paper we present the R package gRc for statistical inference in graphical Gaussian models in which symmetry restrictions have been imposed on the concentration or partial correlation matrix. The models are represented by coloured graphs where parameters associated with edges or vertices of same colour are restricted to being identical. We describe algorithms for maximum likelihood estimation and discuss model selection issues. The paper illustrates the practical use of the gRc package.

Abstract:
The concept of $[r,s,t]$-colourings was recently introduced by Hackmann, Kemnitz and Marangio [3] as follows: Given non-negative integers $r,s$ and $t$, an $[r,s,t]$-colouring of a graph $G=(V(G),E(G))$ is a mapping $c$ from $V(G) \cup E(G)$ to the colour set $\{1,2, \ldots, k\}$ such that $|c(v_i)-c(v_j)| \geq s$ for every two adjacent vertices $v_i$, $v_j$, $|c(e_i)-c(e_j)| \geq s$ for every two adjacent edges $e_i$, $e_j$, and $|c(v_i)-c(e_j)| \geq t$ for all pairs of incident vertices and edges, respectively. The $[r,s,t]$-chromatic number $\chi_{r,s,t}(G)$ of $G$ is defined to be the minimum $k$ such that $G$ admits an [r; s; t]-colouring. In this paper, we determine the $[r,s,t]$-chromatic number for paths.