The List Edge-Coloring Conjecture for Two New Infinite Families of Complete Graphs

Abstract: Let $p$ be an odd prime. We prove the List Edge-Coloring Conjecture for two infinite families of complete graphs: [ χ'{\ell}(K{p-1})=p-2, \qquad χ'{\ell}(K{2p})=2p-1. ] The two proofs control the Pfaffian sign of one-factorizations by complementary modular methods. For $K_{p-1}$, Frobenius and a skew specialization turn Glynn's determinant-coefficient congruence into a squarefree Pfaffian coefficient. A divided difference then reduces the remaining calculation to a single antidiagonal Pfaffian and gives [[x^{\mathbf{1}}]\mbox{Pf}(X)^{p-2}\equiv(-2)^{(p-1)/2}\pmod{p}.] For $K_{2p}$, a weighted Burnside count for the translation group $\Bbb{F}<em>p<sup>2$ isolates a signed cyclic-starter sum. A skew-circulant cofactor identity evaluates its square and gives [ S{2p}\equiv-p\pmod{p^2}. ] In particular, both decisive signed sums are nonzero. Neither congruence is a formal consequence of Latin-square parity: the bipartite determinant sign and the nonbipartite Pfaffian sign are different invariants. Instead, the proofs develop a determinant--Pfaffian bridge and a signed Burnside--Fourier method adapted to the complete-graph sign. We also locate a limit of the latter method. For every even $b\ge4$, the signed trace of a full-support translation on $K_{bp}$ is divisible by $p<sup>b$. For $b=4$ this implies $p^4\mid S_{4p}$ but supplies no nonzero residue, revealing a valuation barrier to the full-support higher-layer argument.