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

TL;DR

This paper establishes improved lower bounds on the average degree of non-complete k-list-critical graphs for k ≥ 6, advancing understanding of their structural properties and extending results to online list-critical graphs.

Contribution

It provides the first improved bounds on average degree for k-list-critical graphs for all k ≥ 6, including online variants, surpassing previous bounds.

Findings

For k ≥ 7, average degree at least k-1 plus a specific positive fraction.

For k=6, average degree at least 5 plus 93/766.

Bounds also apply to online k-list-critical graphs.

Abstract

We improve the best known bounds on average degree of $k$-list-critical graphs for $k \ge 6$. Specifically, for $k \ge 7$ we show that every non-complete $k$-list-critical graph has average degree at least $k-1 + \frac{(k-3)^2 (2 k-3)}{k^4-2 k^3-11 k^2+28 k-14}$ and every non-complete $6$-list-critical graph has average degree at least $5 + \frac{93}{766}$. The same bounds hold for online $k$-list-critical graphs.