Ramsey-Type Results for Path Covers and Path Partitions

A family $\mathcal{P}$ of subgraphs $G$ is called a path cover (resp. partition) if $\bigcup _{P\in \mathcal{P}}V(P)=V(G)$ $\dot\bigcup \mathcal{P}}V(P)=V(G)$) and every element path. The minimum cardinality denoted by ${\rm pc}(G)$ pp}(G)$). In this paper, we characterize the forbidden subgraph conditions assuring us that (or pp}(G)$) bounded constant. Our main results introduce new Ramsey-type problem.