Coloring of a Digraph

TL;DR

This paper introduces a new vertex coloring concept for directed graphs, establishes bounds for the dichromatic number, and characterizes acyclic digraphs using an $L$-matrix, expanding understanding of digraph colorings.

Contribution

It defines the dichromatic number for digraphs, proves it equals one for acyclic digraphs, and introduces sequential coloring notions and a characterization via $L$-matrix.

Findings

Dichromatic number equals 1 for acyclic digraphs.

Bounds and results for the dichromatic number analogous to graph chromatic number.

Introduction of sequential coloring concepts and $L$-matrix characterization.

Abstract

\qquad A \emph{coloring} of a digraph $D=(V,E)$ is a coloring of its vertices following the rule: Let $uv$ be an arc in $D$. If the tail $u$ is colored first, then the head $v$ should receive a color different from that of $u$. The \emph{dichromatic number} $\chi_d(D)$ of $D$ is the minimum number of colors needed in a coloring of $D$. Besides obtaining many results and bounds for $\chi_d(D)$ analogous to that of chromatic number of a graph, we prove $\chi_d(D)=1$ if $D$ is acyclic. New notions of sequential colorings of graphs/digraphs are introduced. A characterization of acyclic digraph is obtained interms of $L$-matrix of a vertex labeled digraph.