11 research outputs found
Oriented coloring on recursively defined digraphs
Coloring is one of the most famous problems in graph theory. The coloring
problem on undirected graphs has been well studied, whereas there are very few
results for coloring problems on directed graphs. An oriented k-coloring of an
oriented graph G=(V,A) is a partition of the vertex set V into k independent
sets such that all the arcs linking two of these subsets have the same
direction. The oriented chromatic number of an oriented graph G is the smallest
k such that G allows an oriented k-coloring. Deciding whether an acyclic
digraph allows an oriented 4-coloring is NP-hard. It follows, that finding the
chromatic number of an oriented graph is an NP-hard problem. This motivates to
consider the problem on oriented co-graphs. After giving several
characterizations for this graph class, we show a linear time algorithm which
computes an optimal oriented coloring for an oriented co-graph. We further
prove how the oriented chromatic number can be computed for the disjoint union
and order composition from the oriented chromatic number of the involved
oriented co-graphs. It turns out that within oriented co-graphs the oriented
chromatic number is equal to the length of a longest oriented path plus one. We
also show that the graph isomorphism problem on oriented co-graphs can be
solved in linear time.Comment: 14 page
Oriented Colouring Graphs of Bounded Degree and Degeneracy
This paper considers upper bounds on the oriented chromatic number, ,
of graphs in terms of their maximum degree and/or their degeneracy
. In particular we show that asymptotically,
where and . This improves a result of MacGillivray, Raspaud, and
Swartz of the form . The rest of the paper is
devoted to improving prior bounds for in terms of and by
refining the asymptotic arguments involved.Comment: 8 pages, 3 figure
On coloring parameters of triangle-free planar -graphs
An -graph is a graph with types of arcs and types of edges. A
homomorphism of an -graph to another -graph is a vertex
mapping that preserves the adjacencies along with their types and directions.
The order of a smallest (with respect to the number of vertices) such is
the -chromatic number of .Moreover, an -relative clique of
an -graph is a vertex subset of for which no two distinct
vertices of get identified under any homomorphism of . The
-relative clique number of , denoted by , is the
maximum such that is an -relative clique of . In practice,
-relative cliques are often used for establishing lower bounds of
-chromatic number of graph families.
Generalizing an open problem posed by Sopena [Discrete Mathematics 2016] in
his latest survey on oriented coloring, Chakroborty, Das, Nandi, Roy and Sen
[Discrete Applied Mathematics 2022] conjectured that for any triangle-free planar -graph and that this
bound is tight for all .In this article, we positively settle
this conjecture by improving the previous upper bound of to , and by
finding examples of triangle-free planar graphs that achieve this bound. As a
consequence of the tightness proof, we also establish a new lower bound of for the -chromatic number for the family of triangle-free
planar graphs.Comment: 22 Pages, 5 figure
Pushable chromatic number of graphs with degree constraints
Pushable homomorphisms and the pushable chromatic number of oriented graphs were introduced by Klostermeyer and MacGillivray in 2004. They notably observed that, for any oriented graph , we have , where denotes the oriented chromatic number of . This stands as first general bounds on . This parameter was further studied in later works.This work is dedicated to the pushable chromatic number of oriented graphs fulfilling particular degree conditions. For all , we first prove that the maximum value of the pushable chromatic number of an oriented graph with maximum degree lies between and which implies an improved bound on the oriented chromatic number of the same family of graphs. For subcubic oriented graphs, that is, when , we then prove that the maximum value of the pushable chromatic number is~ or~. We also prove that the maximum value of the pushable chromatic number of oriented graphs with maximum average degree less than~ lies between~ and~. The former upper bound of~ also holds as an upper bound on the pushable chromatic number of planar oriented graphs with girth at least~
Pushable chromatic number of graphs with degree constraints
International audiencePushable homomorphisms and the pushable chromatic number of oriented graphs were introduced by Klostermeyer and MacGillivray in 2004. They notably observed that, for any oriented graph , we have , where denotes the oriented chromatic number of . This stands as the first general bounds on . This parameter was further studied in later works.This work is dedicated to the pushable chromatic number of oriented graphs fulfilling particular degree conditions. For all , we first prove that the maximum value of the pushable chromatic number of a connected oriented graph with maximum degree lies between and which implies an improved bound on the oriented chromatic number of the same family of graphs. For subcubic oriented graphs, that is, when , we then prove that the maximum value of the pushable chromatic number is~ or~. We also prove that the maximum value of the pushable chromatic number of oriented graphs with maximum average degree less than~ lies between~ and~. The former upper bound of~ also holds as an upper bound on the pushable chromatic number of planar oriented graphs with girth at least~
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