Chromatic Number and Dichromatic Polynomial of Digraphs

TL;DR

This paper extends known graph independence bounds to digraphs, provides a simplified proof for an upper bound on digraph chromatic number, and explores chromatic properties of tournaments, including maximum colorings and chromatic polynomials.

Contribution

It generalizes independence number bounds to digraphs, offers a concise proof for Golowich's chromatic number bound, and characterizes extremal tournaments for coloring and cycle properties.

Findings

Extended independence number bound to digraphs.

Provided a simple proof for Golowich's chromatic bound.

Identified tournaments with extremal coloring and cycle properties.

Abstract

Let $G$ be a graph of order $n$. It is well-known that $\alpha(G)\geq \sum_{i=1}^n \frac{1}{1+d_i}$, where $\alpha(G)$ is the independence number of $G$ and $d_1,\ldots,d_n$ is the degree sequence of $G$. We extend this result to digraphs by showing that if $D$ is a digraph with $n$ vertices, then $ \alpha(D)\geq \sum_{i=1}^n \left( \frac{1}{1+d_i^+} + \frac{1}{1+d_i^-} - \frac{1}{1+d_i}\right)$, where $\alpha(D)$ is the maximum size of an acyclic vertex set of $D$. Golowich proved that for any digraph $D$, $\chi(D)\leq \lceil \frac{4k}{5} \rceil+2$, where $k=max(\Delta^+(D),\Delta^-(D))$. We give a short and simple proof for this result. Next, we investigate the chromatic number of tournaments and determine the unique tournament such that for every integer $k>1$, the number of proper $k$-colorings of that tournament is maximum among all strongly connected tournaments with the same number of vertices. Also, we find the chromatic polynomial of the strongly connected tournament with the minimum number of cycles.