A better lower bound on average degree of 4-list-critical graphs

TL;DR

This paper establishes a new lower bound on the average degree of incomplete k-list-critical graphs, improving previous bounds for k=4,5,6, and extends the result to online k-list-critical graphs, advancing understanding in graph coloring theory.

Contribution

It introduces a tighter lower bound on the average degree of incomplete k-list-critical graphs, enhancing existing bounds for specific values of k and applying to online variants.

Findings

Improved lower bound for k=4,5,6

Bound applies to online k-list-critical graphs

Advances theoretical understanding of list-critical graph properties

Abstract

This short note proves that every incomplete $k$-list-critical graph has average degree at least $k-1 + \frac{k-3}{k^2-2k+2}$. This improves the best known bound for $k = 4,5,6$. The same bound holds for online $k$-list-critical graphs.