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

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.