Degree-dependent Cheeger–bipartiteness bound for Cayley graphons

Determine whether, for every connected Cayley graphon whose parameter \(\mathfrak{d}_W\) is positive, there exists a positive constant \(C\), possibly depending on the degree function of the graphon, such that \(h_W \leq C\beta_W\).

Background

The paper establishes an upper bound of the form $h_W\leq (4/\varepsilon)\beta_W$ for a connected Cayley graphon $W$, under a quantitative positivity assumption on $\mathfrak{d}_W$, where $\mathfrak{d}_W$ measures the gap between the bipartiteness ratio of $W$ and the smallest bipartiteness ratio among index-two measurable subgroups of the underlying compact group. The authors also show that this hypothesis cannot generally be removed: connected non-bipartite Cayley graphons can have unbounded ratios $h_W/\beta_W$ among graphons of fixed degree.

The unresolved issue is whether the quantitative dependence obtained in the theorem can be replaced by some finite constant depending only on the degree function, whenever $\mathfrak{d}_W>0$.