Extremal Cuts of Sparse Random Graphs

For Erdős-Rényi random graphs with average degree γ, and uniformly random γ-regular graph on n vertices, we prove that with high probability the size of both the Max-Cut and maximum bisection are n( γ 4 + P∗ √ γ 4 + o( √ γ)) + o(n) while the size of the minimum bisection is n( γ 4 − P∗ √ γ 4 + o( √ γ)) + o(n). Our derivation relates the free energy of the anti-ferromagnetic Ising model on such graphs to that of the Sherrington-Kirkpatrick model, with P∗ ≈ 0.7632 standing for the ground state energy of the latter, expressed analytically via Parisi’s formula.