The oriented chromatic number of random graphs of bounded degree

TL;DR

This paper investigates the oriented chromatic number of random directed graphs, providing improved bounds and extending classical results from undirected to directed graph models.

Contribution

It introduces new bounds for the oriented chromatic number of random directed graphs, advancing understanding beyond previous undirected graph results.

Findings

Improved upper bound from O(d^2 2^d) to O(√e^d) for the oriented chromatic number.

Extension of classical chromatic number results to directed random graph models.

Enhanced theoretical understanding of coloring properties in directed random graphs.

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 $O(\sqrt{e}^d)$.