Matroid basis graph: Counting Hamiltonian cycles

TL;DR

This paper establishes exponential and super factorial lower bounds on the number of Hamiltonian cycles through any edge in the basis graphs of various matroids, using a unified counting strategy of 4-cycles.

Contribution

It introduces a general method for deriving lower bounds on Hamiltonian cycles in matroid basis graphs, applicable to multiple matroid classes.

Findings

Exponential lower bounds for graphic matroids

Super factorial bounds for generalized Catalan and uniform matroids

A unified counting approach based on 4-cycle enumeration

Abstract

We present exponential and super factorial lower bounds on the number of Hamiltonian cycles passing through any edge of the basis graphs of a graphic, generalized Catalan and uniform matroids. All lower bounds were obtained by a common general strategy based on counting appropriated cycles of length four in the corresponding matroid basis graph.