Necessity of fractional colorability for local-demand bounds

Determine whether the local-demands setting can obtain comparable independent-set marginal guarantees under the weaker assumption that every neighborhood has Hall ratio at most r, rather than the stronger assumption that every neighborhood is fractionally r-colorable.

Background

The paper proves local-demand results for graphs whose neighborhoods are fractionally r-colorable. It contrasts this hypothesis with local-occupancy methods, which can derive related, though weaker, conclusions when neighborhoods have Hall ratio at most r.

The authors suspect that fractional colorability is genuinely necessary for the local-demands framework, but they do not establish this necessity. The open question is therefore whether the stronger fractional-colorability hypothesis can be weakened to bounded Hall ratio.