The niche graphs of multipartite tournaments

The niche graph of a digraph $D$ has $V(D)$ as the vertex set and an edge $uv$ if and only if $(u,w) \in A(D)$ and $(v,w) \in A(D)$, or $(w,u) \in A(D)$ and $(w,v) \in A(D)$ for some $w \in V(D)$. The notion of niche graphs was introduced by Cable et al. [Niche graphs, Discrete Appl. Math. 23 (1989), 231–241] as a variant of competition graphs. If a graph is the niche graph of a digraph $D$, it is said to be niche-realizable through $D$. If a graph $G$ is niche-realizable through a $k$-partite tournament for an integer $k \ge 2$, then we say that the pair $(G, k)$ is niche-realizable. Bowser et al. [Niche graphs and mixed pair graphs of tournaments, J. Graph Theory 31 (1999) 319–332] studied the graphs that are niche-realizable through a tournament and Eoh et al. [The niche graphs of bipartite tournaments, Discrete Appl. Math. 282 (2020) 86–95] recently studied niche-realizable pairs $(G, k)$ for $k=2$. In this paper, we extend their work for $k \ge 3$. We show that the niche graph of a $k$-partite tournament has at most three components if $k \ge 3$ and is connected if $k \ge 4$. Then we find all the niche-realizable pairs $(G, k)$ in each case: $G$ is disconnected; $G$ is a complete graph; $G$ is connected and triangle-free.