Multipartite tournaments in which any two vertices have an $(i,j)$-step common out-neighbor

TL;DR

This paper extends the concept of competitive digraphs to $(i,j)$-step competitive orientations in complete multipartite graphs, providing a general characterization of when such orientations exist.

Contribution

It generalizes previous work by characterizing $(i,j)$-step competitively orientable complete multipartite graphs for all $i,j$, broadening understanding of their structure.

Findings

Provides a complete characterization of $(i,j)$-step competitively orientable graphs.

Extends previous results from the $(1,1)$ case to general $(i,j)$ cases.

Enhances understanding of orientations in multipartite graphs.

Abstract

We say that a digraph $D$ is $(i,j)$-step competitive if any two vertices have an $(i,j)$-step common out-neighbor in $D$ and that a graph $G$ is $(i,j)$-step competitively orientable if there exists an $(i,j)$-step competitive orientation of $G$. In [Choi et al. Competitively orientable complete multipartite graphs. Discrete Mathematics, 345(9):112950, 2022], Choi et al. introduce the notion of competitive digraph and completely characterize competitively orientable complete multipartite graphs in terms of the sizes of its partite sets. Here, a competitive digraph means a $(1,1)$-step competitive digraph. In this paper, the result of Choi et al. has been extended to a general characterization of $(i,j)$-step competitively orientable complete multipartite graphs.