For unweighted graphs, path graphs require at least n-1 global coupling layers, every graph can be compiled in at most 2.5n+2 layers, and perfect matching compilations are tied to Hadamard matrices.
Since each perfect matching can be constructed using n 2 + 1 spin bicliques (one of which is Kn), the entire cycle can be constructed using n + 1 spin bicliques
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Optimizing Cost Hamiltonian Compilation for Max-Cut QAOA on Unweighted Graphs Using Global Controls and Qubit Bit Flips
For unweighted graphs, path graphs require at least n-1 global coupling layers, every graph can be compiled in at most 2.5n+2 layers, and perfect matching compilations are tied to Hadamard matrices.