List Edge-Coloring Conjecture for even-order complete graphs

Prove the List Edge-Coloring Conjecture for every even-order complete graph by establishing that \(\chi'_{\ell}(K_n)=n-1\) whenever \(n\) is even.

Background

The paper states that the List Edge-Coloring Conjecture predicts χℓ′(G)=χ′(G)\chi'_{\ell}(G)=\chi'(G) for every graph. For an even-order complete graph KnK_n, the ordinary chromatic index is n−1n-1, so the conjecture predicts χℓ′(Kn)=n−1\chi'_{\ell}(K_n)=n-1.

The paper proves the conjecture for the two infinite families Kp−1K_{p-1} and K2pK_{2p}, where pp is an odd prime, but explicitly notes that the general even-order complete-graph case remains unresolved.

References

Although such a graph has explicit one-factorizations, list assignments destroy the symmetry used by an ordinary edge coloring. Even this complete-graph case remains open.

— The List Edge-Coloring Conjecture for Two New Infinite Families of Complete Graphs  (2608.22895 - Jafari, 24 Aug 2026) in Section 1, Introduction

For b\ge6, supports k with b/2<k<b are no longer excluded (the first possibility is b=6, k=4). Their traces are not evaluated by the full-support Fourier factorization above. In particular, no general assertion pb\mid S_{bp} follows from the present argument. Proving the List Edge-Coloring Conjecture for these orders would require either an absolute evaluation of a partial-support trace or a different group action.

— The List Edge-Coloring Conjecture for Two New Infinite Families of Complete Graphs  (2608.22895 - Jafari, 24 Aug 2026) in Remark 8.11, Section 8, “A full-support valuation barrier for block translations”