6 research outputs found

    Oriented Colourings of Graphs with Maximum Degree Three and Four

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    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

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    Pushable homomorphisms and the pushable chromatic number χp\chi_p of oriented graphs were introduced by Klostermeyer and MacGillivray in 2004. They notably observed that, for any oriented graph G→\overrightarrow{G}, we have χp(G→)≀χo(G→)≀2χp(G→)\chi_p(\overrightarrow{G}) \leq \chi_o(\overrightarrow{G}) \leq 2 \chi_p(\overrightarrow{G}), where χo(G→)\chi_o(\overrightarrow{G}) denotes the oriented chromatic number of G→\overrightarrow{G}. This stands as first general bounds on χp\chi_p. 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 Δ≄29\Delta \geq 29, we first prove that the maximum value of the pushable chromatic number of an oriented graph with maximum degree Δ\Delta lies between 2Δ2−12^{\frac{\Delta}{2}-1} and (Δ−3)⋅(Δ−1)⋅2Δ−1+2(\Delta-3) \cdot (\Delta-1) \cdot 2^{\Delta-1} + 2 which implies an improved bound on the oriented chromatic number of the same family of graphs. For subcubic oriented graphs, that is, when Δ≀3\Delta \leq 3, we then prove that the maximum value of the pushable chromatic number is~66 or~77. We also prove that the maximum value of the pushable chromatic number of oriented graphs with maximum average degree less than~33 lies between~55 and~66. The former upper bound of~77 also holds as an upper bound on the pushable chromatic number of planar oriented graphs with girth at least~66

    Pushable chromatic number of graphs with degree constraints

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    International audiencePushable homomorphisms and the pushable chromatic number χp\chi_p of oriented graphs were introduced by Klostermeyer and MacGillivray in 2004. They notably observed that, for any oriented graph G→\overrightarrow{G}, we have χp(G→)≀χo(G→)≀2χp(G→)\chi_p(\overrightarrow{G}) \leq \chi_o(\overrightarrow{G}) \leq 2 \chi_p(\overrightarrow{G}), where χo(G→)\chi_o(\overrightarrow{G}) denotes the oriented chromatic number of G→\overrightarrow{G}. This stands as the first general bounds on χp\chi_p. 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 Δ≄29\Delta \geq 29, we first prove that the maximum value of the pushable chromatic number of a connected oriented graph with maximum degree Δ\Delta lies between 2Δ2−12^{\frac{\Delta}{2}-1} and (Δ−3)⋅(Δ−1)⋅2Δ−1+2(\Delta-3) \cdot (\Delta-1) \cdot 2^{\Delta-1} + 2 which implies an improved bound on the oriented chromatic number of the same family of graphs. For subcubic oriented graphs, that is, when Δ≀3\Delta \leq 3, we then prove that the maximum value of the pushable chromatic number is~66 or~77. We also prove that the maximum value of the pushable chromatic number of oriented graphs with maximum average degree less than~33 lies between~55 and~66. The former upper bound of~77 also holds as an upper bound on the pushable chromatic number of planar oriented graphs with girth at least~66
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