Covering 2-Paths with 4-Paths

In this paper, we solve the existence problem for covering the \( 2 \)-paths of \( K_n \) with \( 4 \)-paths. This also settles the spectrum of \( 3 \)-path systems of the line graph of \( K_n \). The proof technique allows the embedding problem for \( (4, 2) \)-path coverings to be settled.