A lower bound on the number of edges in DP-critical graphs

TL;DR

This paper establishes a new lower bound on the minimum number of edges in DP-critical graphs for large enough parameters, improving upon classical bounds and advancing understanding of graph coloring criticality.

Contribution

It provides the first asymptotically better lower bound on the edges of DP-critical graphs compared to classical bounds, for all sufficiently large graphs.

Findings

New lower bound on $f_{DP}(n,k)$ for $k extgreater 4$ and $n extgreater k+1$

Improves bounds on $f_{ ext{list}}(n,k)$ over previous results

First bound surpassing Gallai's classical result for DP-critical graphs

Abstract

A graph $G$ is $k$-critical (list $k$-critical, DP $k$-critical) if $\chi(G)= k$ ($\chi_\ell(G)= k$, $\chi_\mathrm{DP}(G)= k$) and for every proper subgraph $G'$ of $G$, $\chi(G')<k$ ($\chi_\ell(G')< k$, $\chi_\mathrm{DP}(G')<k$). Let $f(n, k)$ ($f_\ell(n, k), f_\mathrm{DP}(n,k)$) denote the minimum number of edges in an $n$-vertex $k$-critical (list $k$-critical, DP $k$-critical) graph. Our main result is that if $k\geq 5$ and $n\geq k+2$, then $$f_\mathrm{DP}(n,k)>\left(k - 1 + \left \lceil \frac{k^2 - 7}{2k-7} \right \rceil^{-1}\right)\frac{n}{2}.$$ This is the first bound on $f_\mathrm{DP}(n,k)$ that is asymptotically better than the well-known bound on $f(n,k)$ by Gallai from 1963. The result also yields a slightly better bound on $f_{\ell}(n,k)$ than the ones known before.