Thin Tree Conjecture

Establish whether, for every real number \(\alpha<1\), there exists an integer \(t\geq 1\) such that every \(t\)-edge-connected graph has a spanning tree that is \(\alpha\)-thin.

Background

The paper defines a set of edges $T$ to be $\alpha$-thin if every nontrivial cut contains at most an $\alpha$-fraction of the edges of that cut in $T$. It then states Goddyn's Thin Tree Conjecture, which asks whether sufficiently high edge connectivity guarantees a spanning tree that is thin for any prescribed $\alpha<1$.

The conjecture is presented as a longstanding problem motivating the study of thin subgraphs and related constructions. The results in the paper establish the existence and abundance of $1/2$-thin subgraphs, but do not resolve the conjecture for spanning trees and arbitrary $\alpha<1$.