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We study partition problems based on two ostensibly different kinds of energyfunctionals defined on $k$-partitions of metric graphs: Cheeger-typefunctionals whose minimisers are the $k$-Cheeger cuts of the graph, and thecorresponding values are the $k$-Cheeger constants of the graph; andfunctionals built using the first eigenvalue of the Laplacian with positive,i.e. absorbing, Robin (delta) vertex conditions at the boundary of thepartition elements. We prove existence of minimising $k$-partitions, $k \geq2$, for both these functionals. We also show that, for each $k \geq 2$, as theRobin parameter $\alpha \to 0$, up to a renormalisation the spectral minimalRobin energy converges to the $k$-Cheeger constant. Moreover, up to asubsequence, the Robin spectral minimal $k$-partitions converge in a naturalsense to a $k$-Cheeger cut of the graph. Finally, we show that as $\alpha \to\infty$ there is convergence in a similar sense to the corresponding Dirichletminimal energy and partitions.  It is strongly expected that similar results hold on general (smooth,bounded) Euclidean domains and manifolds.

Cheeger cuts and Robin spectral minimal partitions of metric graphs