Recovering of the Grassmann graph from the subgraph of non-degenerate subspaces

Abstract

Let ${\mathbb F}$ be a (not necessarily finite) field. A subspace of the vector space ${\mathbb F}^n$ is called {\it non-degenerate} if it is not contained in a coordinate hyperplane. We show that the Grassmann graph of $k$-dimensional subspaces of ${\mathbb F}^n$, $1<k<n-1$, can be recovered from the subgraph of non-degenerate subspaces if $|{\mathbb F}|>n-k$. In the case when ${\mathbb F}={\mathbb F}_q$ is the field of $q$ elements, this subgraph is known as the graph of non-degenerate linear $[n,k]_q$ codes.