Generalized competitively orientable complete multipartite graphs

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. (2022), Choi et al. introduce the notion of the 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.