Buying a Station or Relaxing a Constraint? Spectrum Repacking and Dirac's Conjecture

Spectrum repacking auctions can restore feasibility along two margins: retiring a station via buyout or relaxing a pairwise engineering restriction. I represent stations as vertices in a network and interference restrictions as edges between them, and so station buyouts delete vertices and engineering waivers relax edges. This paper shows that buying out any one station can reduce the number of channels needed, even though relaxing any single interference restriction cannot. The smallest feasible number of channels is $k$ . For every $k\geq4$, I show that there is a network in which removing any station saves one channel, while relaxing any single pairwise restriction does not. This main result follows directly from Dirac’s half-century-old conjecture on vertex-critical graphs, whose chromatic number falls after any vertex is removed. By constructing the missing counterexample at $k=4$ and combining it with earlier results for $k\geq5$, I show that at every $k\geq4$, some $k$-vertex-critical graph has no critical edge, resolving Dirac's conjecture. Market designers auditing feasibility link-by-link systematically can mistake combinatorial complementarity for remedy failure, overlooking station buyouts that unlock the clearing target.