Competitively orientable complete multipartite graphs

TL;DR

This paper characterizes when complete multipartite graphs can be oriented to be competitive digraphs, showing a threshold at size 7 for complete graphs and providing a construction method for such orientations.

Contribution

It provides a complete characterization of competitively orientable complete multipartite graphs based on partite set sizes and offers a construction method for these orientations.

Findings

Complete graphs of order at least 7 are competitively orientable.

A full characterization of competitively orientable complete multipartite graphs is given.

A method to construct competitive multipartite tournaments is provided.

Abstract

We say that a digraph $D$ is competitive if any pair of vertices has a common out-neighbor in $D$ and that a graph $G$ is competitively orientable if there exists a competitive orientation of $G$. The notion of competitive digraphs arose while studying digraph whose competition graphs are complete. We derive some useful properties of competitively orientable graphs and show that a complete graph of order $n$ is competitively orientable if and only if $n \geq 7$. Then we completely characterize a competitively orientable complete multipartite graph in terms of the sizes of its partite sets. Moreover, we present a way to build a competitive multipartite tournament in each of competitively orientable cases.