Set-independence graphs of vector spaces and partial quasigroups

Abstract

As a generalization of independence graphs of vector spaces and groups, we introduce the notions of set-independence graphs of vector spaces and partial quasigroups. The former are characterized for finite-dimensional vector spaces over finite fields. Further, we prove that every finite simple graph is isomorphic to either the independence graph of a partial quasigroup or an induced subgraph of the latter. We also prove that isomorphic partial quasigroups give rise to isomorphic set-independence graphs. As an illustrative example, all finite graphs of order $n\leq 5$ are identified with the independence graph of a partial quasigroup of the same order.