Characterizing paths as mm-step competition graphs

In 2000 Cho, Kim and Nam proved that PnPn, the path on nn vertices, is a 2-step competition graph for all nn. In 2005, Helleloid proved that PnPn is an (n−1)(n−1)- and (n−2)(n−2)-step competition graph for all nn and proved further that of all connected triangle-free graphs on nn vertices, only the star is an mm-step competition graph for m≥nm≥n. In this paper we show that if mm divides n−1n−1 or n−2n−2, then PnPn is an mm-step competition graph and that if n≥6n≥6 and n2≤m≤n−3, then PnPn is not an mm-step competition graph.