Hamiltonian connectivity of some base-cobase graphs

TL;DR

This paper investigates Hamiltonian connectivity in base-cobase graphs of matroids, demonstrating it for certain classes like wheels and whirls, but providing counterexamples with the matroid R_{10}.

Contribution

It extends the understanding of Hamiltonian connectivity from base graphs to base-cobase graphs in specific matroid classes, identifying both positive cases and counterexamples.

Findings

Base-cobase graphs of wheels and whirls are Hamiltonian connected.

Hamiltonian connectivity does not hold universally for all base-cobase graphs, as shown by R_{10}.

Polytopal methods are effective for certain classes but limited for others.

Abstract

There has been wide interest in understanding which properties of base graphs of matroids extend to base-cobase graphs of matroids. A significant result of Naddef and Pulleyblank (1984) shows that the $1$-skeleton of any $(0,1)$-polytope is either a hypercube, or Hamiltonian-connected, i.e. there is a Hamiltonian path connecting any two vertices. In particular, this is true for base graphs of matroids. A natural question raised by Farber, Richter, and Shank (1985) is whether this extends to base-cobase graphs. First, we use the polytopal approach to show Hamiltonian connectivity of base-cobase graphs of series-parallel extensions of lattice path matroids. On the other hand, we show that this method extends to only very special classes related to identically self-dual matroids. Second, we show that base-cobase graphs of wheels and whirls are Hamiltonian connected. Last, we show that the regular matroid $R_{10}$ yields a negative answer to the question of Farber, Richter, and Shank.