Combinatorial Construction of Thin Subsets

Construct thin subsets of graphs by combinatorial methods without relying on eigenvalue arguments, in particular to support progress toward the Thin Tree Conjecture.

Background

The paper notes that several spectral methods construct thin subsets, including the $1/2$-thin subgraphs obtained in the paper by other arguments. However, it identifies an unresolved question concerning whether thin subsets can be constructed combinatorially, without appealing to eigenvalue-based techniques.

This question is specifically motivated by the Thin Tree Conjecture because highly edge-connected graphs need not possess spectrally thin spanning trees. The paper's results provide combinatorial existence results for $1/2$-thin subgraphs, but they do not settle the broader construction problem stated here.

References

We remark that there has been several "spectral" constructions of thin subsets but it remained an open problem whether one can construct thin subsets combinatorially without appealing to eigenvalue arguments (this is specially motivated to address the thin tree conjecture, since t-edge-connected graphs do not necessarily have spectrally thin trees, see ).

— Unweighted Code Sparsifiers and Thin Subgraphs  (2502.02799 - Gharan et al., 5 Feb 2025) in Introduction, immediately after the Thin Tree Conjecture