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University of Victoria

1. Duffy, Christopher. Homomorphisms of (j, k)-mixed graphs.

Degree: Department of Mathematics and Statistics, 2015, University of Victoria

URL: http://hdl.handle.net/1828/6601

A mixed graph is a simple graph in which a subset of the edges have been assigned directions to form arcs. For non-negative integers j and k, a (j, k)−mixed graph is a mixed graph with j types of arcs and k types of edges. The collection of (j, k)−mixed graphs contains simple graphs ((0,1)−mixed graphs), oriented graphs ((1,0)-mixed graphs) and k−edge-coloured graphs ((0, k)−mixed graphs).
A homomorphism is a vertex mapping from one (j,k)−mixed graph to another in which edge type is preserved, and arc type and direction are preserved. An m−colouring of a (j, k)−mixed graph is a homomorphism from that graph to a target with m vertices. The (j, k)−chromatic number of a (j, k)−mixed graph is the least m such that an m−colouring exists. When (j, k) = (0, 1), we see that these definitions are consistent with the usual definitions of graph homomorphism and graph colouring. Similarly, when (j, k) = (1, 0) and (j, k) = (0, k) these definitions are consistent with the usual definitions of homomorphism and colouring for oriented graphs and k−edge-coloured graphs, respectively.
In this thesis we study the (j, k)−chromatic number and related parameters for different families of graphs, focussing particularly on the (1, 0)−chromatic number, more commonly called the oriented chromatic number, and the (0, k)−chromatic number.
In examining oriented graphs, we provide improvements to the upper and lower bounds for the oriented chromatic number of the families of oriented graphs with maximum degree 3 and 4. We generalise the work of Sherk and MacGillivray on the 2−dipath chromatic number, to consider colourings in which vertices at the ends of
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a directed path of length at most k must receive different colours. We examine the implications of the work of Smolikova on simple colourings to study of the oriented chromatic number of the family of oriented planar graphs.
In examining k−edge-coloured graphs we provide improvements to the upper and lower bounds for the family of 2−edge-coloured graphs with maximum degree 3. In doing so, we define the alternating 2−path chromatic number of k−edge-coloured graphs, a parameter similar in spirit to the 2−dipath chromatic number for oriented graphs. We also consider a notion of simple colouring for k−edge-coloured graphs, and show that the methods employed by Smolikova ́ for simple colourings of oriented graphs may be adapted to k−edge-coloured graphs.
In addition to considering vertex colourings, we also consider incidence colourings of both graphs and digraphs. Using systems of distinct representatives, we provide a new characterisation of the incidence chromatic number. We define the oriented incidence chromatic number and find, by way of digraph homomorphism, a connection between the oriented incidence chromatic number and the chromatic number of the underlying graph. This connection motivates our study of the oriented incidence chromatic number of symmetric complete digraphs.
*Advisors/Committee Members: MacGillivray, Gary (supervisor), Sopena, Eric (supervisor).*

Subjects/Keywords: graph theory; coloring; homomorphism; algorithms; complexity

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Duffy, C. (2015). Homomorphisms of (j, k)-mixed graphs. (Thesis). University of Victoria. Retrieved from http://hdl.handle.net/1828/6601

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Duffy, Christopher. “Homomorphisms of (j, k)-mixed graphs.” 2015. Thesis, University of Victoria. Accessed October 19, 2019. http://hdl.handle.net/1828/6601.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Duffy, Christopher. “Homomorphisms of (j, k)-mixed graphs.” 2015. Web. 19 Oct 2019.

Vancouver:

Duffy C. Homomorphisms of (j, k)-mixed graphs. [Internet] [Thesis]. University of Victoria; 2015. [cited 2019 Oct 19]. Available from: http://hdl.handle.net/1828/6601.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Duffy C. Homomorphisms of (j, k)-mixed graphs. [Thesis]. University of Victoria; 2015. Available from: http://hdl.handle.net/1828/6601

Not specified: Masters Thesis or Doctoral Dissertation

2. Duffy, Christopher. Homomorphisms of (j,k)-mixed graphs : Homomorphisms of (j,k)-mixed graphs.

Degree: Docteur es, Informatique, 2015, Bordeaux; University of Victoria

URL: http://www.theses.fr/2015BORD0128

Un graphe mixte est un graphe simple tel que un sous-ensemble des arêtes a une orientation. Pour entiers non négatifs j et k, un graphe mixte-(j,k) est un graphe mixte avec j types des arcs and k types des arêtes. La famille de graphes mixte-(j,k) contient graphes simple, (graphes mixte−(0,1)), graphes orienté (graphes mixte−(1,0)) and graphe coloré arête −k (graphes mixte−(0,k)).Un homomorphisme est un application sommet entre graphes mixte−(j,k) que tel les types des arêtes sont conservés et les types des arcs et leurs directions sont conservés. Le nombre chromatique−(j,k) d’un graphe mixte−(j,k) est le moins entier m tel qu’il existe un homomorphisme à une cible avec m sommets. Quand on observe le cas de (j,k) = (0,1), on peut déterminer ces définitions correspondent à les définitions usuel pour les graphes.Dans ce mémoire on etude le nombre chromatique−(j,k) et des paramètres similaires pour diverses familles des graphes. Aussi on etude les coloration incidence pour graphes and digraphs. On utilise systèmes de représentants distincts et donne une nouvelle caractérisation du nombre chromatique incidence. On define le nombre chromatique incidence orienté et trouves un connexion entre le nombre chromatique incidence orienté et le nombre chromatic du graphe sous-jacent.

A mixed graph is a simple graph in which a subset of the edges have been assigned directions to form arcs. For non-negative integers j and k, a (j,k)−mixed graph is a mixed graph with j types of arcs and k types of edges. The collection of (j,k)−mixed graphs contains simple graphs ((0,1)−mixed graphs), oriented graphs ((1,0)−mixed graphs) and k−edge- coloured graphs ((0,k)−mixed graphs).A homomorphism is a vertex mapping from one (j,k)−mixed graph to another in which edge type is preserved, and arc type and direction are preserved. The (j,k)−chromatic number of a (j,k)−mixed graph is the least m such that an m−colouring exists. When (j,k)=(0,1), we see that these definitions are consistent with the usual definitions of graph homomorphism and graph colouring.In this thesis we study the (j,k)−chromatic number and related parameters for different families of graphs, focussing particularly on the (1,0)−chromatic number, more commonly called the oriented chromatic number, and the (0,k)−chromatic number.In addition to considering vertex colourings, we also consider incidence colourings of both graphs and digraphs. Using systems of distinct representatives, we provide a new characterisation of the incidence chromatic number. We define the oriented incidence chromatic number and find, by way of digraph homomorphism, a connection between the oriented incidence chromatic number and the chromatic number of the underlying graph. This connection motivates our study of the oriented incidence chromatic number of symmetric complete digraphs.

Subjects/Keywords: Graphe; Homomorphisme; Graphe orienté; Graphe orientée coloration; Graph; Homomorphism; Oriented Graphs; Oriented colouring

Record Details Similar Records

❌

APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Duffy, C. (2015). Homomorphisms of (j,k)-mixed graphs : Homomorphisms of (j,k)-mixed graphs. (Doctoral Dissertation). Bordeaux; University of Victoria. Retrieved from http://www.theses.fr/2015BORD0128

Chicago Manual of Style (16^{th} Edition):

Duffy, Christopher. “Homomorphisms of (j,k)-mixed graphs : Homomorphisms of (j,k)-mixed graphs.” 2015. Doctoral Dissertation, Bordeaux; University of Victoria. Accessed October 19, 2019. http://www.theses.fr/2015BORD0128.

MLA Handbook (7^{th} Edition):

Duffy, Christopher. “Homomorphisms of (j,k)-mixed graphs : Homomorphisms of (j,k)-mixed graphs.” 2015. Web. 19 Oct 2019.

Vancouver:

Duffy C. Homomorphisms of (j,k)-mixed graphs : Homomorphisms of (j,k)-mixed graphs. [Internet] [Doctoral dissertation]. Bordeaux; University of Victoria; 2015. [cited 2019 Oct 19]. Available from: http://www.theses.fr/2015BORD0128.

Council of Science Editors:

Duffy C. Homomorphisms of (j,k)-mixed graphs : Homomorphisms of (j,k)-mixed graphs. [Doctoral Dissertation]. Bordeaux; University of Victoria; 2015. Available from: http://www.theses.fr/2015BORD0128