The strong thin tree conjecture

Theorem 1.1: there is an absolute $C>0$ such that every finite loopless $k$-edge-connected multigraph with at least two vertices has a spanning tree $T$ with $|\delta_T(S)|\le(C/k)|\delta_G(S)|$ for all cuts. The proof extracts tree packings on low-resistance edges and iterates a Marcus-Spielman-Srivastava sparsification. The companion gives a deterministic polynomial-time construction, polynomial in the binary input length. Neither paper gives an explicit constant or a new ATSP approximation ratio (constant-factor ATSP was already known by other methods), and spectrally thin trees are not claimed.

Theorem 1.1: there is an absolute $C>0$ such that every finite loopless $k$-edge-connected multigraph with at least two vertices has a spanning tree $T$ with $|\delta_T(S)|\le(C/k)|\delta_G(S)|$ for all cuts. The proof extracts tree packings on low-resistance edges and iterates a Marcus-Spielman-Srivastava sparsification. The companion gives a deterministic polynomial-time construction, polynomial in the binary input length. Neither paper gives an explicit constant or a new ATSP approximation ratio (constant-factor ATSP was already known by other methods), and spectrally thin trees are not claimed.