Edge Lower Bounds for List Critical Graphs, via Discharging

TL;DR

This paper improves the lower bounds on the number of edges in $k$-list-critical graphs using a simplified and modular discharging method, advancing understanding of graph coloring complexity.

Contribution

It introduces a new, simpler discharging approach to establish tighter lower bounds on edges in $k$-list-critical graphs, enhancing previous results.

Findings

Improved lower bounds on edges in $k$-list-critical graphs.

Discharging method simplifies previous proofs.

Results contribute to graph coloring theory.

Abstract

A graph $G$ is $k$-critical if $G$ is not $(k-1)$-colorable, but every proper subgraph of $G$ is $(k-1)$-colorable. A graph $G$ is $k$-choosable if $G$ has an $L$-coloring from every list assignment $L$ with $|L(v)|=k$ for all $v$, and a graph $G$ is \emph{$k$-list-critical} if $G$ is not $(k-1)$-choosable, but every proper subgraph of $G$ is $(k-1)$-choosable. The problem of bounding (from below) the number of edges in a $k$-critical graph has been widely studied, starting with work of Gallai and culminating with the seminal results of Kostochka and Yancey, who essentially solved the problem. In this paper, we improve the best lower bound on the number of edges in a $k$-list-critical graph. Our proof uses the discharging method, which makes it simpler and more modular than previous work in this area.