Oriented Colouring Graphs of Bounded Degree and Degeneracy

TL;DR

This paper improves an upper bound of MacGillivray, Raspaud, and Swartz of the form $\chi_o(G) \leq 2^{\chi_2(G)} -1$ to a polynomial upper bound for graphs with bounded $2$-dipath chromatic number and gives an improved asymptotic upper Bound for all graphs with maximum degree at most $\Delta".

Abstract

This paper considers upper bounds on the oriented chromatic number $χ_o(G)$, of an oriented graph $G$ in terms of its $2$-dipath chromatic number $χ_2(G)$, degeneracy $d(G)$, and maximum degree $Δ(G)$. In particular, we show that for all graphs $G$ with $χ_2(G) \leq k$ where $k \geq 2$ and $d(G) \leq t$ where $t \geq \log_2(k)$, $χ_o(G) = 33/10(k t^2 2^t)$. This improves an upper bound of MacGillivray, Raspaud, and Swartz of the form $χ_o(G) \leq 2^{χ_2(G)} -1$ to a polynomial upper bound for many classes of graphs, in particular, those with bounded degeneracy. Additionally, we asymptotically improve bounds for the oriented chromatic number in terms of maximum degree and degeneracy. For instance, we show that $χ_o(G) \leq (2\ln2 +o(1))Δ^2 2^Δ$ for all graphs, and $χ_o(G) \leq (2+o(1))Δd 2^d$ for graphs where degeneracy grows sublinearly in maximum degree. Here the asypmtotics are in $Δ$. The former improves the asymptotics of a results by Kostochka, Sopena, and Zhu \cite{kostochka1997acyclic}, while the latter improves the asymptotics of a result by Aravind and Subramanian \cite{aravind2009forbidden}. Both improvements are by a constant factor.

Figures (5)

• Figure 1: Three examples of oriented colourings (consider each component as its own graph).
• Figure 2: An example of an oriented homomorphism. Colours are depicted as numbers following the definition, as well as actual colours.
• Figure 3: A triangle and a directed subdivision of a triangle where the new edges all form $2$-dipaths.
• Figure 4: A digraph with two components is depicted, along with a $2$-dipath colouring which is not an oriented colouring.
• Figure 5: The Cayley graph $T:= Cayley(\mathbb{Z}/7\mathbb{Z};\{1,2,4\})$ which is a smallest $(2,1)$-comprehensive graph.

Theorems & Definitions (24)

• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• Lemma 3.4
• proof
• Theorem 3.5
• proof
• Lemma 3.6