Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The dib-chromatic number of digraphs

Nahid Javier-Nol, Christian Rubio-Montiel, Ingrid Torres-Ramos

Abstract

We study an extension to directed graphs of the parameter called the $b$-chromatic number of a graph in terms of acyclic vertex colorings: the dib-chromatic number. We give general bounds for this parameter. We also show some results about tournaments and regular digraphs.

The dib-chromatic number of digraphs

Abstract

We study an extension to directed graphs of the parameter called the -chromatic number of a graph in terms of acyclic vertex colorings: the dib-chromatic number. We give general bounds for this parameter. We also show some results about tournaments and regular digraphs.

Paper Structure

This paper contains 5 sections, 15 theorems, 17 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Bounds
  3. On tournaments
  4. On regular digraphs
  5. Statements and Declarations

Key Result

Theorem 1

Let $D$ be a digraph of order $n$. Assume that the vertices $u_1,\dots ,u_n$ of $D$ are ordered such that $deg^+(u_1)\geq deg^+(u_2) \geq \dots \geq deg^+(u_n)$. Let $t^+(D) := \max \{i \colon deg^+(u_i) \geq i-1\}$ be the maximum number $i$ such that $D$ contains at least $i$ vertices of out-degree

Figures (1)

  • Figure 1: For each digraph $D$, $dib(D)=2$.

Theorems & Definitions (27)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Corollary 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • ...and 17 more