Regular graphs are graphs in which all vertices have the same degree. Many properties of these graphs are known. Such graphs play an important role in modeling network configurations where equipment limitations impose a restriction on the maximum number of links emanating from a node. These limitations do not enforce strict regularity, and it becomes interesting to investigate nonregular graphs that are in some sense close to regular. This dissertation explores a particular class of almost regular graphs in detail and defines generalizations on this class. A linear-time algorithm for the creation of arbitrarily large graphs of the discussed class is provided, and a polynomial-time algorithm for recognizing graphs in the class is given. Several invariants for the class are discussed. The edge-face chromatic number χef of a plane graph G is the minimum number of colors that must be assigned to the edges and faces of G such that no edge or face of G receives the same color as an edge or face with which it is incident or adjacent. A well-known result for the upper bound of χef exists for graphs with maximum degree Δ ≥ 10. We present a tight upper bound for plane graphs with Δ = 9.

The Internet of Things (IoT) is an important and expanding technology used for a large variety of applications to monitor and automate processes. The aim of this thesis is to present a way to find and assign locally unique IDs to access points (APs) in enormous wireless IoT systems where mobile tags are traversing the network and communicating with multiple APs simultaneously. This is done in order to improve the robustness of the system and increase the battery time of the tags. The resulting algorithm is based on transforming the problem into a graph coloring problem and solving it using approximate methods. Two metaheuristics: Simulated annealing and tabu search were implemented and compared for this purpose. Both of these showed similar results and neither was clearly superior to the other. Furthermore, the presented algorithm can also exclude nodes from the coloring based on the results in order to ensure a proper solution that also satisfies a robustness criterion. A metric was also created in order for a user to intuitively evaluate the quality of a given solution. The algorithm was tested and evaluated on a system of 222 APs for which it produced good results.

Hadwiger's Conjecture from 1943 states that every graph with no Kt minor is (t-1)-colorable; it remains wide open for t ≥ 7. For positive integers t and s, let Kt-s denote the family of graphs obtained from the complete graph Kt by removing s edges. We say that a graph has no Kt-s minor if it has no H minor for every H in Kt-s. In 1971, Jakobsen proved that every graph with no K7-2 minor is 6-colorable. In this dissertation, we first study the extremal functions for K8-4 minors, K9-6 minors, and K10-12 minors. We show that every graph on n ≥ 9 vertices with at least 4.5n-12 edges has a K8-4 minor, every graph on n ≥ 9 vertices with at least 5n-14 edges has a K9-6 minor, and every graph on n ≥ 10 vertices with at least 5.5n-17.5 edges has a K10-12 minor. We then prove that every graph with no K8-4 minor is 7-colorable, every graph with no K9-6 minor is 8-colorable, and every graph with no K10-12 minor is 9-colorable. The proofs use the extremal functions as well as generalized Kempe chains of contraction-critical graphs obtained by Rolek and Song and a method for finding minors from three different clique subgraphs, originally developed by Robertson, Seymour, and Thomas in 1993 to prove Hadwiger's Conjecture for t = 6. Our main results imply that H-Hadwiger's Conjecture is true for each graph H on 8 vertices that is a subgraph of every graph in K8-4, each graph H on 9 vertices that is a subgraph of every graph in K9-6, and each graph H on 10 vertices that is a subgraph of every graph in K10-12.

ENGLISH ABSTRACT: Determining the (classical) chromatic number of a graph (i.e. finding the smallest number of colours with which the vertices of a graph may be coloured so that no two adjacent vertices receive the same colour) is a well known combinatorial optimization problem and is widely encountered in scheduling problems. Since the late 1960s the notion of the chromatic number has been generalized in several ways by relaxing the restriction of independence of the colour classes. / AFRIKAANSE OPSOMMING: Die bepaling van die (klassieke) chromatiese getal van ’n grafiek (naamlik die kleinste aantal kleure waarmee die punte van ’n grafiek gekleur kan word sodat geen twee naasliggende punte dieselfde kleur ontvang nie) is ’n bekende kombinatoriese optimeringsprobleem wat wyd in skeduleringstoepassings te¨egekom word. Sedert die laat 1960s is die definisie van die chromatiese getal op verskeie maniere veralgemeen deur die vereiste van onafhanklikheid van die kleurklasse te verslap. / Thesis (DPhil)--Stellenbosch University, 2007.

Vertex-distinguishing edge-colorings (vdec colorings) are a restriction of proper edge-colorings. These special colorings require that the sets of edge colors incident to every vertex be distinct. This is a relatively new field of study. We present a survey of known results concerning vdec colorings. We also define a new matrix which may be used to study vdec colorings, and examine its properties. We find several bounds on the eigenvalues of this matrix, as well as results concerning its determinant, and other properties. We finish by examining related topics and open problems.

The famous Four Color Theorem states that any planar graph can be properly colored using at most four colors. However, if we want to properly color the square of a planar graph (or alternatively, color the graph using distinct colors on vertices at distance up to two from each other), we will always require at least \Delta + 1 colors, where \Delta is the maximum degree in the graph. For all \Delta, Wegner constructed planar graphs (even without 3-cycles) that require about \frac{3}{2} \Delta colors for such a coloring. To prove a stronger upper bound, we consider only planar graphs that contain no 4-cycles and no 5-cycles (but which may contain 3-cycles). Zhu, Lu, Wang, and Chen showed that for a graph G in this class with \Delta \ge 9, we can color G^2 using no more than \Delta + 5 colors. In this thesis we improve this result, showing that for a planar graph G with maximum degree \Delta \ge 32 having no 4-cycles and no 5-cycles, at most \Delta + 3 colors are needed to properly color G^2. Our approach uses the discharging method, and the result extends to list-coloring and other related coloring concepts as well.

A 3-map is a 3-region colorable map. They have been studied by Craft and White in their paper 3-maps. This thesis introduces topological graph theory and then investigates 3-maps in detail, including examples, special types of 3-maps, the use of 3-maps to find the genus of special graphs, and a generalization known as n-maps.

Thesis (M.Sc.)--Memorial University of Newfoundland, 2005. / Bibliography: leaves 78-83.

Vertex-distinguishing edge-colorings (vdec colorings) are a restriction of proper edge-colorings. These special colorings require that the sets of edge colors incident to every vertex be distinct. This is a relatively new field of study. We present a survey of known results concerning vdec colorings. We also define a new matrix which may be used to study vdec colorings, and examine its properties. We find several bounds on the eigenvalues of this matrix, as well as results concerning its determinant, and other properties. We finish by examining related topics and open problems.

The work in this thesis can be divided into two different parts. In the first part, we suggest an approximate edge 3-coloring polynomial time algorithm for cubic graphs. For any cubic graph with n vertices, using this coloring algorithm, we get an edge 3-coloring with at most n/3 error vertices. In the second part, we study Jim Propp's Rotor-Router model on some non-bipartite graph. We find the difference between the number of chips at vertices after performing a walk on this graph using Propp model and the expected number of chips after a random walk. It is known that for line of integers and d-dimenional grid, this deviation is constant. However, it is also proved that for k-ary infinite trees, for some initial configuration the deviation is no longer a constant and say it is D. We present a similar study on some non-bipartite graph constructed from k-ary infinite trees and conclude that for this graph with the same initial configuration, the deviation is almost (k²)D.