The niche graphs of multipartite tournaments

TL;DR

This paper investigates the properties of niche graphs derived from multipartite tournaments, extending previous work to cases where the underlying graph has at least three parts, and classifies niche-realizable pairs for various graph types.

Contribution

It extends the characterization of niche-realizable pairs to k-partite tournaments with k ≥ 3, including classifications for disconnected, complete, and triangle-free graphs.

Findings

Niche graph of a k-partite tournament has at most three components for k ≥ 3.

Such graphs are connected if k ≥ 4.

Complete and triangle-free graphs' niche-realizability is fully characterized.

Abstract

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 graph was introduced by Cable et al. (1989) as a variant of competition graph. 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. (1999) studied the graphs that are niche-realizable through a tournament and Eoh et al. (2018) studied niche-realizable pairs $(G, k)$ for $k=2$. In this paper, we study niche-realizable pairs $(G, k)$ when $G$ is a graph and $k$ is an integer at least $3$ to extend their work. 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 $(G, k)$ when $G$ is a disconnected graph, when $G$ is a complete graph, and when $G$ is a connected triangle-free graph.