Note on Hamiltonicity of basis graphs of even delta-matroids

TL;DR

This paper proves that basis graphs of even delta-matroids are Hamiltonian under broad conditions, extending classical results for matroids and providing new insights into their cycle structures.

Contribution

It establishes Hamiltonicity of basis graphs of even delta-matroids and characterizes their cycle properties, generalizing known matroid results with new proofs.

Findings

Basis graphs of even delta-matroids are Hamiltonian if they have more than two vertices.

Vertices and edges in these graphs belong to cycles of all sufficiently large lengths.

If not a hypercube, each vertex and edge belongs to cycles of every length above certain thresholds.

Abstract

We show that the basis graph of an even delta-matroid is Hamiltonian if it has more than two vertices. More strongly, we prove that for two distinct edges $e$ and $f$ sharing a common end, it has a Hamiltonian cycle using $e$ and avoiding $f$ unless it has at most two vertices or it is a cycle of length at most four. We also prove that if the basis graph is not a hypercube graph, then each vertex belongs to cycles of every length $\ell\ge 3$, and each edge belongs to cycles of every length $\ell \ge 4$. For the last theorem, we provide two proofs, one of which uses the result of Naddef (1984) on polytopes and the result of Chepoi (2007) on basis graphs of even delta-matroids, and the other is a direct proof using various properties of even delta-matroids. Our theorems generalize the analogous results for matroids by Holzmann and Harary (1972) and Bondy and Ingleton (1976).