Article in volume
Authors:
Title:
The niche graphs of multipartite tournaments
PDFSource:
Discussiones Mathematicae Graph Theory 43(4) (2023) 1123-1146
Received: 2020-11-23 , Revised: 2021-06-29 , Accepted: 2021-07-01 , Available online: 2021-07-21 , https://doi.org/10.7151/dmgt.2424
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 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.
Keywords:
niche graph, multipartite tournament, niche-realizable pair, true twins, triangle-free graph
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