Minimum Number of Arcs in k-Critical Digraphs with Order at Most 2k-1

Core Concepts

The minimum number of arcs in k-critical digraphs with order at most 2k-1 is 2(n/2 - (p^2 + 1)), where n = k + p and 2 ≤ p ≤ k-1.

Abstract

The paper investigates the minimum number of arcs in k-critical digraphs, which are digraphs where every proper subdigraph has a dichromatic number less than k.
The main result is that if n = k + p, where 2 ≤ p ≤ k-1, then the minimum number of arcs in a k-critical digraph of order n is 2(n/2 - (p^2 + 1)). The authors also provide an exact characterization of the k-critical digraphs that achieve this minimum.
The proof proceeds by induction on k, considering several cases based on the structure of the k-critical digraph. Key results used include Gallai's theorems on the structure of critical graphs, as well as recent work on the minimum number of arcs in critical digraphs.
The paper also discusses some consequences of the main result, including a characterization of k-critical digraphs on k+2 vertices and bounds on the number of isomorphism types of k-critical digraphs on k+p vertices.

Stats

n = k + p, where 2 ≤ p ≤ k-1
-→ext(k, n) = 2(n/2 - (p^2 + 1))

Quotes

"Let n, k and p be integers with n = k + p and 2 ≤ p ≤ k-1, then -→ext(k, n) = 2(n/2 - (p^2 + 1)), and we give an exact characterisation of k-critical digraphs for which equality holds."

Key Insights Distilled From

Minimum number of arcs in $k$-critical digraphs with order at most $2k-1$

by Lucas Picasa... at arxiv.org 04-30-2024

https://arxiv.org/pdf/2310.03584.pdf

Deeper Inquiries

How can the results in this paper be extended to k-critical digraphs with order greater than 2k-1?

The results in the paper provide insights into the minimum number of arcs in k-critical digraphs with order at most 2k-1. To extend these results to k-critical digraphs with order greater than 2k-1, one could explore the relationship between the dichromatic number and the structure of larger digraphs. By analyzing the critical properties of digraphs with higher orders, researchers can potentially derive new bounds or characterizations for the minimum number of arcs in these larger k-critical digraphs. Additionally, studying the decomposable and indecomposable critical digraphs in the context of higher-order digraphs could offer valuable insights into the extremal functions and the structure of k-critical digraphs with orders exceeding 2k-1.

What are the implications of this work for the computational complexity of determining the dichromatic number of a given digraph?

The work on k-critical digraphs and their minimum number of arcs has implications for the computational complexity of determining the dichromatic number of a given digraph. The dichromatic number of a digraph is a fundamental concept in graph theory, and finding an efficient algorithm to compute it is a challenging problem. The results in the paper, particularly the characterization of k-critical digraphs and the extremal functions, provide valuable insights into the structure of digraphs with specific coloring properties. By understanding the critical properties of digraphs, researchers can potentially develop more efficient algorithms for determining the dichromatic number. The insights gained from studying k-critical digraphs can lead to the development of heuristic approaches or exact algorithms that leverage the structural properties identified in the paper to compute the dichromatic number of a given digraph more effectively.

Are there any connections between the structure of k-critical digraphs and other areas of graph theory or computer science?

The structure of k-critical digraphs, as studied in the paper, has connections to various areas of graph theory and computer science.

Graph Coloring: The concept of k-critical digraphs is closely related to graph coloring theory. Understanding the critical properties of digraphs can provide insights into the chromatic number and dichromatic number of graphs, which are essential topics in graph coloring.

Complexity Theory: The study of k-critical digraphs and their extremal functions can have implications for complexity theory. By analyzing the computational complexity of determining the dichromatic number in k-critical digraphs, researchers can gain insights into the complexity of related graph problems.

Algorithm Design: The results on k-critical digraphs can inspire the development of new algorithms for graph coloring and related optimization problems. The structural properties identified in k-critical digraphs can be leveraged to design efficient algorithms for various graph-related tasks.

Overall, the study of k-critical digraphs contributes to a deeper understanding of graph theory and its applications in computer science, offering connections to diverse areas within the field.