  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).
