Colourings of oriented connected cubic graphs

doi:10.1016/j.disc.2020.112021

Discrete Mathematics

Volume 343, Issue 10, October 2020, 112021

https://doi.org/10.1016/j.disc.2020.112021 Get rights and content

Abstract

In this note we show every orientation of a connected cubic graph admits an oriented 8-colouring. This lowers the best-known upper bound for the chromatic number of the family of orientations of connected cubic graphs. We further show that every such oriented graph admits a 2-dipath 7-colouring. These results imply that either the oriented chromatic number for the family of orientations of connected cubic graphs equals the 2-dipath chromatic number or the long-standing conjecture of Sopena (Sopena, 1997) regarding the chromatic number of orientations of connected cubic graphs is false.

Section snippets

Introduction and preliminary notions

An oriented graph is a simple graph equipped with an orientation of its edges as arcs. Equivalently, an oriented graph is an antisymmetric loopless digraph. We call an oriented graph $G$ properly subcubic when its underlying simple graph, denoted $U (G)$ , has maximum degree three and a vertex of degree at most two. The degree of a vertex in $G$ is its degree in $U (G)$ . We call an oriented graph $G$ connected when $U (G)$ is connected. For $u v, v w \in A (G)$ , we say $u v w$ is a $2$ -dipath; $v$ is between $u$ and $w$ ; and $v$ is

Oriented colourings of orientations of connected cubic graphs

For convenience we provide the following result regarding $Q R_{7}$ .

Lemma 2

[6]

(1)
The tournament $Q R_{7}$ is vertex transitive and arc transitive.
(2)
For $y z \in A (Q R_{7})$ , there exists a pair of distinct vertices $x, x^{'} \in V (Q R_{7})$ so that $x y z$ and $x^{'} y z$ are directed cycles in $Q R_{7}$ .

We begin with technical lemma.

Lemma 3

If $G$ is an oriented cubic graph with no source and no sink, then

•
$G$ contains a vertex with out-degree $2$ whose out-neighbours both have in-degree $2$ ; or
•
$G$ contains a vertex of out-degree $2$ that has an out-neighbour of in-degree $2$ and an

$2$ -dipath Colourings of Orientations of Cubic Graphs

For an oriented graph $G$ , let $G^{2}$ be the simple undirected graph formed from $G$ by first adding an edge between any pair of vertices at directed distance exactly $2$ in $G$ (i.e., vertices at the end of an induced $2$ -dipath) and then changing all arcs to edges. One easily observes $χ (G^{2}) = χ_{2 d} (G)$ . Thus we approach our study of $χ_{2 d} (F_{3})$ by examining the chromatic number of graphs of the form $G^{2}$ for $G \in F_{3}$ . Let $F_{3}^{2} = {G^{2} | G \in F_{3}}$ . In [2] the authors establish $ω (F_{3}^{2}) = 7$ . Thus $χ_{2 d} (F_{3}) \geq 7$ . Here we show $χ_{2 d} (F_{3}) = 7$ .

Lemma 8

If $G$ is

Discussion

In an early investigation into the oriented chromatic number of orientations of connected cubic graphs Sopena [8] conjectured $χ_{o} (F_{3}^{C}) = 7$ . This conjecture has been verified for all orientations of connected cubic graphs with fewer than $20$ vertices [7]. Here we have shown $χ_{o} (F_{3}^{C}) \in {7, 8}$ and $χ_{2 d} (F_{3}) = 7$ . Our results imply exactly one of the following must be true:

(1)
$χ_{o} (F_{3}^{C}) = 8$ or
(2)
$χ_{o} (F_{3}^{C}) = χ_{2 d} (F_{3}) = 7$ .

In other words, either Sopena’s conjecture is false or any orientation of a cubic graph with $2$ -dipath

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Cited by (0)

¹: Research supported by the Natural Science and Research Council of Canada .

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NoteColourings of oriented connected cubic graphs

Abstract

Section snippets

Introduction and preliminary notions

Oriented colourings of orientations of connected cubic graphs

[6]

2-dipath Colourings of Orientations of Cubic Graphs

Discussion

Declaration of Competing Interest

Discrete Math.

Discrete Math.

Discrete Math.

Graph Theory

Oriented coloring of graphs with low maximum degree

Discrete Math.

Note
Colourings of oriented connected cubic graphs

$2$ -dipath Colourings of Orientations of Cubic Graphs