Reconstruct Hamilton-cycle counts from linearly sized induced subgraphs

Establish whether there exist constants 0 ≤ c < 1 and n_0 ≥ 0 such that, for every integer n ≥ n_0, the number of Hamilton cycles in every n-vertex graph G is uniquely determined by the multiset of induced subgraphs of G on ⌊cn⌋ vertices.

Background

The paper establishes reconstruction results for matching polynomials and F-tiling polynomials, which encode counts of highly disconnected spanning structures. It then proposes that analogous reconstruction may hold for connected spanning structures, specifically Hamilton cycles.

The authors present this as an explicit conjecture. A supporting result of Tutte shows that Hamilton-cycle counts can be reconstructed from the deck of induced subgraphs on n−1 vertices, but the conjecture asks whether a fixed linear fraction cn with c < 1 suffices for all sufficiently large n.