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

Let $p$ be an odd prime. We prove the List Edge-Coloring Conjecture for two infinite families of complete graphs: $${\chi }_{\mathrm{\ell }}^{\mathrm{\prime }}(K_{p-1})=p-2,{\chi }_{\mathrm{\ell }}^{\mathrm{\prime }}(K_{2p})=2p-1.$$ 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 $$[x^{\mathbf{1}}]Pf(X{)}^{p-2}\equiv (-2{)}^{(p-1)\mathrm{/}2}(\mathrm{m}\mathrm{o}\mathrm{d}p)\mathrm{.}$$ For $K_{2p}$, a weighted Burnside count for the translation group ${\mathbb{F}}_p^2$ isolates a signed cyclic-starter sum. A skew-circulant cofactor identity evaluates its square and gives $$S_{2p}\equiv -p(\mathrm{m}\mathrm{o}\mathrm{d}{p}^2)\mathrm{.}$$ 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 $b\ge 4$, the signed trace of a full-support translation on $K_{bp}$ is divisible by $p^b$. 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.