On the Holt-Klee property for oriented matroid programming

The Holt-Klee theorem says that the graph of a d-polytope, with edges oriented by a linear function on P that is not constant on any edge, admits d independent monotone paths from the source to the sink. We prove that the digraphs obtained from oriented matroid programs of rank d + 1 on n + 2 elements, which include those from d-polytopes with n facets, admit d independent monotone paths from source to sink if d <= 4. This was previously only known to hold for d <= 3 and n <= 6. (C) 2021 Elsevier Ltd. All rights reserved.