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

A graph $G$ is $k$-critical (list $k$-critical, DP $k$-critical) if $χ(G)= k$ ($χ_\ell(G)= k$, $χ_\mathrm{DP}(G)= k$) and for every proper subgraph $G’$ of $G$, $χ(G’)<k$ ($χ_\ell(G’)< k$, $χ_\mathrm{DP}(G’)<k$). 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.

💡 Research Summary

The paper investigates the sparsity of DP‑critical graphs, establishing a new lower bound on the minimum number of edges in an n‑vertex, k‑critical graph under the DP‑coloring framework. DP‑coloring, introduced by Dvořák and Postle, generalizes list‑coloring by representing each vertex’s list of permissible colors as a set of vertices in a covering multigraph H, with edges between these sets forming matchings that encode adjacency constraints. The DP‑chromatic number χ₍DP₎(G) is the smallest integer k such that every cover (H, L) with |L(v)| ≥ k admits an (H, L)‑coloring. A graph is DP‑k‑critical if χ₍DP₎(G)=k but every proper subgraph has DP‑chromatic number at most k − 1.

Historically, lower bounds on the edge count of k‑critical graphs have been studied for ordinary coloring (χ) and list‑coloring (χ_ℓ). Dirac’s classic bound (k − 1)n/2 + (k − 3)/2 and Gallai’s stronger bound (k − 1 + (k − 3)/(k² − 3))·n/2 have served as benchmarks. Subsequent improvements by Krivelevich, Kostochka–Stiebitz, and Rabern refined these constants, but all pertain to ordinary or list coloring. For DP‑coloring, only the Dirac‑type bound was known (Bernshteyn and Kostochka showed the same (k − 1)n/2 + (k − 3)/2 holds for DP‑critical graphs), leaving a gap in the asymptotic regime where Gallai’s bound is superior.

The authors close this gap by proving that for any integers k ≥ 5 and n ≥ k + 2, \