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Cheeger vs. Dirichlet Cheeger equality on trees

Determine whether, for every tree graph G and every k, the standard k-way Cheeger constant equals the k-way Dirichlet Cheeger constant; equivalently, prove or refute h_k(G)=H_k(G) for all k on trees.

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Background

The authors introduce Dirichlet Cheeger constants H_k(G) based on subsets with Dirichlet boundary, and show general inequalities relating H_k and the k-way Cheeger constants h_k. They verify equality for paths and star graphs.

Extending equality to all trees would unify two multiway isoperimetric notions on a broad and important class of graphs and has implications for variational 1-Laplacian eigenvalues and min–max/max–min frameworks.

References

It can be verified that eq_cheeger_dirichlet_inequality is actually an equality for path graphs and star graphs. Thus we have the following (strong) open problem on tree graphs. Let $G$ be a tree graph, is it true or false that the $k$-way Cheeger constants introduced in \Cref{DEf:Cheeger_constant} are equal to the $k$-way Dirichlet Cheeger constants introduced in \Cref{DEF:dirichlet_cheeger_constants}, i.e. $h_k(G)= H_k(G),\;\; \forall k?$}]}

eq_cheeger_dirichlet_inequality:

Hk(G)=max⁡∣A∣=N−k+1h1(A)≤hk(G).H_{k}(G) = \max_{|A|=N-k+1}h_1(A) \leq h_k(G).

— Nonlinear spectral graph theory (2504.03566 - Deidda et al., 4 Apr 2025) in Subsection 6.2, “Relations with Dirichlet Cheeger constants”