Completing the Boundary Case of the Mahmoodian–Mirzakhani Conjecture and 117 New Computational 5-Cycle Decompositions of Complete Tripartite Graphs

Let K_{r,s,t}, with r ≤ s ≤ t, denote the complete tripartite graph whose partite sets have sizes r, s, t. Mahmoodian and Mirzakhani gave three necessary conditions for K_{r,s,t} to admit a decomposition into 5-cycles and conjectured that these conditions are sufficient. One of the conditions is t ≤ 4rs/(r + s). We prove the conjecture for every odd triple on the extremal boundary t = 4rs/(r + s).

The proof is constructive. After reducing an arbitrary odd boundary triple to (r, s, t) = (hga, hgb, hab), a + b = 4g, we give an explicit cyclic decomposition of K_{ga,gb,ab} and use the Mahmoodian and Mirzakhani scaling theorem to supply the common factor h. Together with the previously known all-even result, this settles the conjecture for every triple satisfying the boundary condition with equality.

We also report explicit computer-generated C5-decompositions for 117 odd triples satisfying the necessary conditions, including many strict-interior cases. To the best of our knowledge, after comparing these triples with the results and constructions in the previously available literature cited in this paper, all 117 cases were previously unresolved: no decomposition for any of them had been reported, and none of the 117 triples is covered by those earlier existence results, constructions, or their recursive consequences. Moreover, according to our comparison with the previous literature, these 117 certificates together with the boundary construction settle every previously unresolved triple satisfying the necessary conditions with fewer than 4400 edges; equivalently, to the best of our knowledge, no such unresolved case remains below this edge threshold. Each computation is supplied as a machine-readable cycle-list certificate and can be checked independently by a short Python verifier. We describe the exact-cover search used to obtain the certificates, list all 117 triples and their forced cycle-type counts, and give a complete human-readable edge-label-matrix certificate for K_{9,19,23}. The certificates, auxiliary data files, verifier, and a copy of this paper are archived in this Zenodo record.