7 research outputs found
Oriented Colourings of Graphs with Maximum Degree Three and Four
We show that any orientation of a graph with maximum degree three has an
oriented 9-colouring, and that any orientation of a graph with maximum degree
four has an oriented 69-colouring. These results improve the best known upper
bounds of 11 and 80, respectively
On the oriented chromatic number of dense graphs
Let be a graph with vertices, edges, average degree , and maximum degree . The \emph{oriented chromatic number} of is the maximum, taken over all orientations of , of the minimum number of colours in a proper vertex colouring such that between every pair of colour classes all edges have the same orientation. We investigate the oriented chromatic number of graphs, such as the hypercube, for which . We prove that every such graph has oriented chromatic number at least . In the case that , this lower bound is improved to . Through a simple connection with harmonious colourings, we prove a general upper bound of \Oh{\Delta\sqrt{n}} on the oriented chromatic number. Moreover this bound is best possible for certain graphs. These lower and upper bounds are particularly close when is ()-regular for some constant , in which case the oriented chromatic number is between and
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)âmixed graph is a mixed graph with j types of arcs and k types of edges. The collection of (j,k)âmixed graphs contains simple graphs ((0,1)âmixed graphs), oriented graphs ((1,0)âmixed graphs) and kâedge- coloured graphs ((0,k)âmixed graphs).A homomorphism is a vertex mapping from one (j,k)âmixed graph to another in which edge type is preserved, and arc type and direction are preserved. The (j,k)âchromatic number of a (j,k)âmixed graph is the least m such that an mâcolouring 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)âchromatic number and related parameters for different families of graphs, focussing particularly on the (1,0)âchromatic number, more commonly called the oriented chromatic number, and the (0,k)âchromatic 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 arĂȘtes a une orientation. Pour entiers non nĂ©gatifs j et k, un graphe mixte-(j,k) est un graphe mixte avec j types des arcs and k types des arĂȘtes. La famille de graphes mixte-(j,k) contient graphes simple, (graphes mixteâ(0,1)), graphes orientĂ© (graphes mixteâ(1,0)) and graphe colorĂ© arĂȘte âk (graphes mixteâ(0,k)).Un homomorphisme est un application sommet entre graphes mixteâ(j,k) que tel les types des arĂȘtes sont conservĂ©s et les types des arcs et leurs directions sont conservĂ©s. Le nombre chromatiqueâ(j,k) dâun graphe mixteâ(j,k) est le moins entier m tel quâil existe un homomorphisme Ă une cible avec m sommets. Quand on observe le cas de (j,k) = (0,1), on peut dĂ©terminer ces dĂ©finitions correspondent Ă les dĂ©finitions usuel pour les graphes.Dans ce mĂ©moire on etude le nombre chromatiqueâ(j,k) et des paramĂštres similaires pour diverses familles des graphes. Aussi on etude les coloration incidence pour graphes and digraphs. On utilise systĂšmes de reprĂ©sentants distincts et donne une nouvelle caractĂ©risation du nombre chromatique incidence. On define le nombre chromatique incidence orientĂ© et trouves un connexion entre le nombre chromatique incidence orientĂ© et le nombre chromatic du graphe sous-jacent
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
Une contribution à la théorie d' homomorphisme et de coloration des graphes
An oriented graph is a directed graph with no cycle of length at most two. A homomorphism of an oriented graph to another oriented graph is an arc preserving vertex mapping. To push a vertex is to switch the direction of the arcs incident to it. An orientable graph is an equivalence class of oriented graph with respect to the push operation. An orientable graph [ââG] admits a homomorphism to an orientable graph [ââH] if an element of [ââG] admits a homomorphism to an element of [ââH]. A signified graph (G, ÎŁ) is a graph whose edges are assigned either a positive sign or a negative sign, while ÎŁ denotes the set of edges with negative signs assigned to them. A homomorphism of a signified graph to another signified graph is a vertex mapping such that the image of a positive edge is a positive edge and the image of a negative edge is a negative edge. A signed graph [G, ÎŁ] admits a homomorphism to a signed graph [H, Î] if an element of [G, ÎŁ] admits a homomorphism to an element of [H, Î]. The oriented chromatic number of an oriented graph ââG is the minimum order of an oriented graph ââH such that ââG admits a homomorphism to ââH. A set R of vertices of an oriented graph ââG is an oriented relative clique if no two vertices of R can have the same image under any homomorphism. The oriented relative clique number of an oriented graph ââG is the maximum order of an oriented relative clique of ââG. An oriented clique or an oclique is an oriented graph whose oriented chromatic number is equal to its order. The oriented absolute clique number of an oriented graph ââG is the maximum order of an oclique contained in ââG as a subgraph. The chromatic number, the relative chromatic number and the absolute chromatic number for orientable graphs, signified graphs and signed graphs are defined similarly. In this thesis we study the chromatic number, the relative clique number and the absolute clique number of the above mentioned four types of graphs. We specifically study these three parameters for the family of outerplanar graphs, of outerplanar graphs with given girth, of planar graphs and of planar graphs with given girth. We also try to investigate the relation between the four types of graphs and prove some results regarding that. In this thesis, we provide tight bounds for the absolute clique number of these families in all these four settings. We provide improved bounds for relative clique numbers for the same. For some of the cases we manage to provide improved bounds for the chromatic number as well. One of the most difficult results that we prove here is that the oriented absolute clique number of the family of planar graphs is at most 15. This result settles a conjecture made by Klostermeyer and MacGillivray in 2003. Using the same technique we manage to prove similar results for orientable planar graphs and signified planar graphs. We also prove that the signed chromatic number of triangle-free planar graphs is at most 25 using the discharging method. This also implies that the signified chromatic number of trianglefree planar graphs is at most 50 improving the previous upper bound. We also study the 2-dipath and oriented L(p, q)-labeling (labeling with a condition for distance one and two) for several families of planar graphs. It was not known if the categorical product of orientable graphs and of signed graphs exists. We prove both the existence and also provide formulas to construct them. Finally, we propose some conjectures and mention some future directions of works to conclude the thesis.Dans cette thĂšse, nous considĂ©rons des questions relatives aux homomorphismes de quatre types distincts de graphes : les graphes orientĂ©s, les graphes orientables, les graphes 2-arĂȘte colorĂ©s et les graphes signĂ©s. Pour chacun des ces quatre types, nous cherchons Ă dĂ©terminer le nombre chromatique, le nombre de clique relatif et le nombre de clique absolu pour diffĂ©rentes familles de graphes planaires : les graphes planaires extĂ©rieurs, les graphes planaires extĂ©rieurs de maille fixĂ©e, les graphes planaires et les graphes planaires de maille fixĂ©e. Nous Ă©tudions Ă©galement les Ă©tiquetages "2-dipath" et "L(p,q)" des graphes orientĂ©s et considĂ©rons les catĂ©gories des graphes orientables et des graphes signĂ©s. Nous Ă©tudions enfin les diffĂ©rentes relations pouvant exister entre ces quatre types d'homomorphismes de graphes
A contribution to the theory of graph homomorphisms and colorings
Dans cette thĂšse, nous considĂ©rons des questions relatives aux homomorphismes de quatre types distincts de graphes : les graphes orientĂ©s, les graphes orientables, les graphes 2-arĂȘte colorĂ©s et les graphes signĂ©s. Pour chacun des ces quatre types, nous cherchons Ă dĂ©terminer le nombre chromatique, le nombre de clique relatif et le nombre de clique absolu pour diffĂ©rentes familles de graphes planaires : les graphes planaires extĂ©rieurs, les graphes planaires extĂ©rieurs de maille fixĂ©e, les graphes planaires et les graphes planaires de maille fixĂ©e. Nous Ă©tudions Ă©galement les Ă©tiquetages "2-dipath" et "L(p,q)" des graphes orientĂ©s et considĂ©rons les catĂ©gories des graphes orientables et des graphes signĂ©s. Nous Ă©tudions enfin les diffĂ©rentes relations pouvant exister entre ces quatre types d'homomorphismes de graphes.An oriented graph is a directed graph with no cycle of length at most two. A homomorphism of an oriented graph to another oriented graph is an arc preserving vertex mapping. To push a vertex is to switch the direction of the arcs incident to it. An orientable graph is an equivalence class of oriented graph with respect to the push operation. An orientable graph [ââG] admits a homomorphism to an orientable graph [ââH] if an element of [ââG] admits a homomorphism to an element of [ââH]. A signified graph (G, ÎŁ) is a graph whose edges are assigned either a positive sign or a negative sign, while ÎŁ denotes the set of edges with negative signs assigned to them. A homomorphism of a signified graph to another signified graph is a vertex mapping such that the image of a positive edge is a positive edge and the image of a negative edge is a negative edge. A signed graph [G, ÎŁ] admits a homomorphism to a signed graph [H, Î] if an element of [G, ÎŁ] admits a homomorphism to an element of [H, Î]. The oriented chromatic number of an oriented graph ââG is the minimum order of an oriented graph ââH such that ââG admits a homomorphism to ââH. A set R of vertices of an oriented graph ââG is an oriented relative clique if no two vertices of R can have the same image under any homomorphism. The oriented relative clique number of an oriented graph ââG is the maximum order of an oriented relative clique of ââG. An oriented clique or an oclique is an oriented graph whose oriented chromatic number is equal to its order. The oriented absolute clique number of an oriented graph ââG is the maximum order of an oclique contained in ââG as a subgraph. The chromatic number, the relative chromatic number and the absolute chromatic number for orientable graphs, signified graphs and signed graphs are defined similarly. In this thesis we study the chromatic number, the relative clique number and the absolute clique number of the above mentioned four types of graphs. We specifically study these three parameters for the family of outerplanar graphs, of outerplanar graphs with given girth, of planar graphs and of planar graphs with given girth. We also try to investigate the relation between the four types of graphs and prove some results regarding that. In this thesis, we provide tight bounds for the absolute clique number of these families in all these four settings. We provide improved bounds for relative clique numbers for the same. For some of the cases we manage to provide improved bounds for the chromatic number as well. One of the most difficult results that we prove here is that the oriented absolute clique number of the family of planar graphs is at most 15. This result settles a conjecture made by Klostermeyer and MacGillivray in 2003. Using the same technique we manage to prove similar results for orientable planar graphs and signified planar graphs. We also prove that the signed chromatic number of triangle-free planar graphs is at most 25 using the discharging method. This also implies that the signified chromatic number of trianglefree planar graphs is at most 50 improving the previous upper bound. We also study the 2-dipath and oriented L(p, q)-labeling (labeling with a condition for distance one and two) for several families of planar graphs. It was not known if the categorical product of orientable graphs and of signed graphs exists. We prove both the existence and also provide formulas to construct them. Finally, we propose some conjectures and mention some future directions of works to conclude the thesis