The oriented chromatic number of random graphs of bounded degree

TL;DR

The oriented chromatic number of the directed models $\vec{G}(n,p)$ and $\vec{\mathcal{G}} (n,d)$ is considered, improving the best known upper bound from $O(d^2 2^d$ to $O(\sqrt{e}^d)$.

Abstract

The chromatic number of the random graph $\mathcal{G}(n,p)$ has long been studied and has inspired several landmark results. In the case where $p = d/n$, Achlioptas and Naor showed the chromatic number is asymptotically concentrated at $k_d$ or $k_d+1$, where $k_d$ is the smallest integer such that $d < 2k_d\log k_d$. Kemkes et al. later proved the same result holds for $\mathcal{G}(n,d)$, the random $d$-regular graph. We consider the oriented chromatic number of the directed models $\vec{\mathcal{G}}(n,p)$ and $\vec{\mathcal{G}}(n,d)$, improving the best known upper bound from $O(d^2 2^d)$ to $O(\sqrt{e}^d)$.

Figures (2)

• Figure 1: Small tournaments for colouring oriented $2$-regular graphs
• Figure 2: Oriented $5$-cycle with $\chi_o(C) = 4$

Theorems & Definitions (34)

• Definition 2.3
• proof
• Definition 3.1
• proof
• proof
• Definition 3.4
• proof
• Remark 3.6
• proof
• proof