Triangle-free $m$-step competition graphs.

Given a digraph $D$ and a positive number $m$, the $m$-step competition graph of $D$ is defined to have the same vertex set as $D$ and an edge joining two distinct vertices $u$ and $v$ if and only if there exist a $(u,w)$-directed walk and a $(v,w)$-directed walk both having length $m$ for some vertex $w$ in $D$. We call a graph an $m$-step competition graph if it is the $m$-step competition graph of a digraph. The notion of $m$-step competition graph was introduced by Cho \emph{et al.} \cite{cho2000m} as one of the variants of competition graph which was introduced by Cohen \cite{Cohen} while studying predator-prey concepts in ecological food webs. In this paper, we first extend a result given by Helleloid \cite{helleloid2005connected} stating that for all positive integers $m \geq n$, the only connected triangle-free $m$-step competition graph on $n$ vertices is the star graph. We show that the result is true for arbitrary positive integer $m \geq 2$ as long as it is the $m$-step competition graph of a digraph having a source. We go further to completely characterize the digraphs each of whose weak components has a source and whose $m$-step competition graphs are triangle-free for some integer $m \ge 2$. We also compute the number of digraphs with a source whose $m$-step competition graphs are connected and triangle-free for some integer $m \ge 2$.