
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 is a coloring of its vertices following the rule: Let be an arc in . If the tail is colored first, then the head should receive a color different from that of . The \emph{dichromatic number} of is the minimum number of colors needed in a coloring of . Besides obtaining many results and bounds for analogous to that of chromatic number of a graph, we prove if is acyclic. New notions of sequential colorings of graphs/digraphs are introduced. A characterization of acyclic digraph is obtained interms of -matrix of a vertex labeled digraph.
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Taxonomy
TopicsGraph Labeling and Dimension Problems · Advanced Graph Theory Research · graph theory and CDMA systems
