A Study of kk-dipath Colourings of Oriented Graphs

We examine $t$-colourings of oriented graphs in which, for a fixed integer $k \geq 1$, vertices joined by a directed path of length at most $k$ must be assigned different colours. A homomorphism model that extends the ideas of Sherk for the case $k=2$ is described. Dichotomy theorems for the complexity of the problem of deciding, for fixed $k$ and $t$, whether there exists such a $t$-colouring are proved.Comment: 14 page

Oriented Colouring Graphs of Bounded Degree and Degeneracy

This paper considers upper bounds on the oriented chromatic number, $\chi_o$, of graphs in terms of their maximum degree $\Delta$ and/or their degeneracy $d$. In particular we show that asymptotically, $\chi_o \leq \chi_2 f(d) 2^d$ where $f(d) \geq (\frac{1}{\log_2(e) -1} + \epsilon) d^2$ and $\chi_2 \leq 2^{\frac{f(d)}{d}}$. This improves a result of MacGillivray, Raspaud, and Swartz of the form $\chi_o \leq 2^{\chi_2} -1$. The rest of the paper is devoted to improving prior bounds for $\chi_o$ in terms of $\Delta$ and $d$ by refining the asymptotic arguments involved.Comment: 8 pages, 3 figure

On the existence and non-existence of improper homomorphisms of oriented and 22-edge-coloured graphs to reflexive targets

We consider non-trivial homomorphisms to reflexive oriented graphs in which some pair of adjacent vertices have the same image. Using a notion of convexity for oriented graphs, we study those oriented graphs that do not admit such homomorphisms. We fully classify those oriented graphs with tree-width $2$ that do not admit such homomorphisms and show that it is NP-complete to decide if a graph admits an orientation that does not admit such homomorphisms. We prove analogous results for $2$-edge-coloured graphs. We apply our results on oriented graphs to provide a new tool in the study of chromatic number of orientations of planar graphs -- a long-standing open problem

Homomorphisms of (j,k)-mixed graphs

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.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

Homomorphically Full Oriented Graphs

Homomorphically full graphs are those for which every homomorphic image is isomorphic to a subgraph. We extend the definition of homomorphically full to oriented graphs in two different ways. For the first of these, we show that homomorphically full oriented graphs arise as quasi-transitive orientations of homomorphically full graphs. This in turn yields an efficient recognition and construction algorithms for these homomorphically full oriented graphs. For the second one, we show that the related recognition problem is GI-hard, and that the problem of deciding if a graph admits a homomorphically full orientation is NP-complete. In doing so we show the problem of deciding if two given oriented cliques are isomorphic is GI-complete

On oriented cliques with respect to push operation

International audienceAn oriented graph is a directed graph without any directed cycle of length at most 2. An oriented clique is an oriented graph whose non-adjacent vertices are connected by a directed 2-path. To push a vertex v of a directed graph G is to change the orientations of all the arcs incident to v. A push clique is an oriented clique that remains an oriented clique even if one pushes any set of vertices of it. We show that it is NP-complete to decide if an undirected graph is the underlying graph of a push clique or not. We also prove that a planar push clique can have at most 8 vertices and provide an exhaustive list of planar push cliques

The Edmonds-Giles Conjecture and its Relaxations

Given a directed graph, a directed cut is a cut with all arcs oriented in the same direction, and a directed join is a set of arcs which intersects every directed cut at least once. Edmonds and Giles conjectured for all weighted directed graphs, the minimum weight of a directed cut is equal to the maximum size of a packing of directed joins. Unfortunately, the conjecture is false; a counterexample was first given by Schrijver. However its ”dual” statement, that the minimum weight of a dijoin is equal to the maximum number of dicuts in a packing, was shown to be true by Luchessi and Younger. Various relaxations of the conjecture have been considered; Woodall’s conjecture remains open, which asks the same question for unweighted directed graphs, and Edmond- Giles conjecture was shown to be true in the special case of source-sink connected directed graphs. Following these inquries, this thesis explores different relaxations of the Edmond- Giles conjecture