6 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
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~