Ramsey-type results for path covers and path partitions. II. Digraphs

TL;DR

This paper extends Ramsey-type results to directed graphs, establishing bounds on path cover and partition numbers based on forbidden substructures, thus advancing understanding of digraph path decompositions.

Contribution

It introduces conditions involving forbidden structures in digraphs that guarantee bounded path cover and partition numbers, expanding prior graph results to directed graphs.

Findings

Identifies forbidden structures that bound path cover/partition numbers in digraphs.

Establishes relationships between structural constraints and path decompositions.

Provides theoretical bounds for path cover and partition numbers in weakly connected digraphs.

Abstract

Recently, the authors gave Ramsey-type results for the path cover/partition number of graphs. In this paper, we continue the research about them focusing on digraphs, and find a relationship between the path cover/partition number and forbidden structures in digraphs. Let $D$ be a weakly connected digraph. A family $\mathcal{P}$ of subdigraphs of $D$ is called a {\it path cover} (resp. a {\it path partition}) of $D$ if $\bigcup _{P\in \mathcal{P}}V(P)=V(D)$ (resp. $\dot\bigcup _{P\in \mathcal{P}}V(P)=V(D)$) and every element of $\mathcal{P}$ is a directed path. The minimum cardinality of a path cover (resp. a path partition) of $D$ is denoted by ${\rm pc}(D)$ (resp. ${\rm pp}(D)$). In this paper, we find forbidden structure conditions assuring us that ${\rm pc}(D)$ (or ${\rm pp}(D)$) is bounded by a constant.