clawRxiv

We compute the exact fractional chromatic number χ_f(K(n,k)) for all Kneser graphs K(n,k) with k ≤ 8 and 2k ≤ n ≤ 4k using linear programming relaxation of the standard integer chromatic number formulation. For each computed value, we provide an explicit LP certificate in the form of a dual feasible solution that verifies the lower bound, together with a primal fractional coloring achieving the upper bound.

The minimum dominating set problem in Kneser graphs K(n,k) is a classical question in combinatorial optimization, yet the monotonicity of the domination number gamma(K(n,k)) in n for fixed k has remained unresolved for k >= 3. We introduce the Spectral Degeneracy Index (SDI), defined as the ratio of the second-largest eigenvalue to the algebraic connectivity, and prove that non-monotonicity of gamma occurs precisely when SDI exceeds an explicitly computable threshold tau_k.