Oriented Colouring Graphs of Bounded Degree and Degeneracy

TL;DR

This paper establishes new polynomial upper bounds on the oriented chromatic number of graphs based on their degeneracy, maximum degree, and 2-dipath chromatic number, improving previous exponential bounds.

Contribution

It introduces polynomial bounds on the oriented chromatic number for graphs with bounded degeneracy and degree, refining existing exponential bounds and asymptotic results.

Findings

Polynomial upper bound $rac{33}{10}k t^2 2^t$ for graphs with $ ext{chi}_2(G) extless= k$ and degeneracy $ extless= t$

Asymptotic bound $ extless= (2 ext{ln}2 + o(1)) ext{Delta}^2 2^ ext{Delta}$ for maximum degree $ ext{Delta}$

Asymptotic bound $ extless= (2+o(1)) ext{Delta} d 2^d$ for graphs with degeneracy $d$ growing sublinearly in $ ext{Delta}$

Abstract

This paper considers upper bounds on the oriented chromatic number $\chi_o(G)$, of an oriented graph $G$ in terms of its $2$-dipath chromatic number $\chi_2(G)$, degeneracy $d(G)$, and maximum degree $\Delta(G)$. In particular, we show that for all graphs $G$ with $\chi_2(G) \leq k$ where $k \geq 2$ and $d(G) \leq t$ where $t \geq \log_2(k)$, $\chi_o(G) = 33/10(k t^2 2^t)$. This 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 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 $\chi_o(G) \leq (2\ln2 +o(1))\Delta^2 2^\Delta$ for all graphs, and $\chi_o(G) \leq (2+o(1))\Delta d 2^d$ for graphs where degeneracy grows sublinearly in maximum degree. Here the asypmtotics are in $\Delta$. 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.