Combinatorial tiling conjecture for paired non-acyclic graphs

Let $G=(V,E)$ be a non-acyclic graph, with a fixed-point-free involution $\mu:V\to V$ and, for every vertex $v$, a bijection $\sigma_v:\delta(v)\to\delta(\mu(v))$. Assume that $\lambda(v,\mu(v))=\deg(v)$ and $\sigma_{\mu(v)}=\sigma_v^{-1}$. Tiling conjecture. There exists a nonempty list of cycles of $G$ such that, for every pair of edges $e$ and $f$ incident with a vertex $v$, the number of listed cycles containing both $e$ and $f$ equals the number containing both $\sigma_v(e)$ and $\sigma_v(f)$. If $G$ has a connected component with at least four vertices, the list may moreover be required to contain a cycle of length greater than two. This is the combinatorial restatement of the paper’s tiling conjecture and is intended to provide the graph-theoretic structure needed for polygonal surface constructions.