Proof of the List Edge Coloring Conjecture for Complete Graphs of Prime Degree

Uwe Schauz

doi:10.37236/4084

Abstract

We prove that the list-chromatic index and paintability index of $K_{p+1}$ is $p$, for all odd primes $p$. This implies that the List Edge Coloring Conjecture holds for complete graphs with less then 10 vertices. It also shows that there are arbitrarily big complete graphs for which the conjecture holds, even among the complete graphs of class 1. Our proof combines the Quantitative Combinatorial Nullstellensatz with the Paintability Nullstellensatz and a group action on symmetric Latin squares. It displays various ways of using different Nullstellensätze. We also obtain a partial proof of a version of Alon and Tarsi's Conjecture about even and odd Latin squares.

Highlights

IntroductionFor complete graphs of class 2 (odd n) this was proven by Haggkvist and Janssen, who presented the following upper bound for all n ∈ Z+ in [HaJa] (the generalization to paintability can be found in [Sch4]): Theorem 1.2
Given a graph G = (V, E) and a nonempty list Le for every edge e ∈ E, we setL := e∈E Le and say that G is L-list-edge colorable or edge L-choosable if it is possibleL to assign to each edge e ∈ E a color from the list Le in such a way that any two adjacent edges of G receive different colors
We prove that the list-chromatic index and paintability index of Kp+1 is p, for all odd primes p

Summary

Introduction

For complete graphs of class 2 (odd n) this was proven by Haggkvist and Janssen, who presented the following upper bound for all n ∈ Z+ in [HaJa] (the generalization to paintability can be found in [Sch4]): Theorem 1.2. We will only have to count the symmetric Latin squares that are fixed under that group action We found this crucial trick in [Dr] (see [StWa]), where Drisko used it the electronic journal of combinatorics 21(3) (2014), #P3.43 to give a partial proof of Alon and Tarsi’s Conjecture about even and odd Latin squares. Theorem 4.2 will yield, in combination with Theorem 3.1, our main results, Theorem 1.4 and Theorem 1.5

The Polynomial Approach

Interpreting the Coefficient

A group action on Latin squares