Characterizing paths as -step competition graphs

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