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On the dib-chromatic number of a digraph

Juan José Montellano-Ballesteros, Christian Rubio-Montiel

TL;DR

We address the problem of defining and understanding the dib-chromatic number $\text{dib}(D)$ for digraphs, the maximum $k$ for which an acyclic $b$-coloring with $k$ colors exists. The main approach proves existence by showing $dc(D) \le \text{dib}(D)$ via $b$-irreducible colorings, and then analyzes dib$(D)$ in bipartite settings to derive bounds such as $\text{dib}(D) \le n+m-\beta(D)+1$ and $\text{dib}(D) \le n+1$ for connected bipartite graphs with $|A|=n\le m=|B|$, while identifying cases where $\text{dib}(D) > 2$ and determining $\text{dib}(D)=3$ for simple, $2$-regular bipartite digraphs. The paper also presents open problems, including relaxing degree conditions to obtain $\text{dib}(D) \ge 3$ and exploring links to the digrundy/dac framework in tournaments. Overall, it advances the theory of acyclic colorings in digraphs and provides a foundation for further algorithmic and combinatorial investigations in directed coloring.

Abstract

An acyclic coloring of a digraph that maximizes the number of colors such that each color class has a vertex pointing to all other classes and a vertex pointing to it from all other classes is known as the dib-chromatic number of a digraph. In this paper, we answer the question about the existence of the dib-chromatic number and study the dib-chromatic number of bipartite digraphs.

On the dib-chromatic number of a digraph

TL;DR

We address the problem of defining and understanding the dib-chromatic number for digraphs, the maximum for which an acyclic -coloring with colors exists. The main approach proves existence by showing via -irreducible colorings, and then analyzes dib in bipartite settings to derive bounds such as and for connected bipartite graphs with , while identifying cases where and determining for simple, -regular bipartite digraphs. The paper also presents open problems, including relaxing degree conditions to obtain and exploring links to the digrundy/dac framework in tournaments. Overall, it advances the theory of acyclic colorings in digraphs and provides a foundation for further algorithmic and combinatorial investigations in directed coloring.

Abstract

An acyclic coloring of a digraph that maximizes the number of colors such that each color class has a vertex pointing to all other classes and a vertex pointing to it from all other classes is known as the dib-chromatic number of a digraph. In this paper, we answer the question about the existence of the dib-chromatic number and study the dib-chromatic number of bipartite digraphs.

Paper Structure

This paper contains 6 sections, 16 theorems, 8 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Existence
  3. Bipartite digraphs
  4. On simple bipartite digraphs
  5. Proposed problems
  6. Statements and Declarations

Key Result

Lemma 1

Let $D$ be a digraph and $\Gamma: V(D)\rightarrow [r]$ be an acyclic coloring of $D$ using $r$ colors. If $\Gamma$ is $b$-irreducible, then $\Gamma$ is a $b$-coloring of $D$.

Figures (1)

  • Figure 1: There are bipartite graphs with 3 components whose $\text{dib}(D)$ is 2.

Theorems & Definitions (46)

  • Lemma 1
  • proof
  • Claim 2
  • Claim 3
  • Theorem 4
  • proof
  • Proposition 5
  • proof
  • Corollary 6
  • Lemma 7
  • ...and 36 more