Hyperplanes and hamiltonian circuits in Perfect Matroid Designs with fixed basis

TL;DR

This paper investigates the enumeration of Hamiltonian circuits containing a fixed basis and hyperplanes excluding a fixed basis in perfect matroid designs, with applications to finite geometries, and provides algorithms for their identification.

Contribution

It introduces methods to count and find Hamiltonian circuits and hyperplanes in perfect matroid designs, especially in finite geometries, advancing understanding of their combinatorial structure.

Findings

Algorithms for finding hyperplanes and Hamiltonian circuits.

Enumeration results for specific matroid classes.

Applications to projective and affine geometries.

Abstract

We study the number of hamiltonian circuits, containing a fixed basis, and the number of hyperplanes, which do not contain a fixed basis in perfect matroid designs. Projective and affine finite geometries are considered as examples of such matroids. We give algorithms to find the hyperplanes and the hamiltonian circuits in such cases.