The List Edge-Coloring Conjecture for New Infinite Families

Abstract

The List Edge-Coloring Conjecture predicts that any graph whose edges can be colored with $k$ colors can also be colored from arbitrary lists of $k$ colors. We prove its stronger online form for two new infinite families, $K_{p-1}$ and $K_{2p}$, where $p$ is an odd prime. For even $n$, order the vertices of $K_n$ and draw each perfect matching as arcs above them. Count crossings separately within each matching, and let $S_n$ be the number of decompositions into perfect matchings having an even total crossing count minus the number having an odd total. Then \[ S_{p-1}\equiv\left(\frac{-2}{p}\right)\pmod p, \qquad S_{2p}\equiv-p\pmod {p^2}. \] The two congruences are governed by the same elementary matching sum over $\F_p$, although their proofs use the prime $p$ differently. Their nonzero residues give the conjectured values even in the online game. They also treat the corresponding complete graphs with one perfect matching removed, as well as $K_{2p}$ after deleting some, but not all, of a natural cyclic family of $p$ disjoint perfect matchings.