Structure in sparse $k$-critical graphs

TL;DR

This paper advances understanding of the structure and density of $k$-critical graphs, proving new bounds that incorporate the number of disjoint $K_{k-1}$ and $K_{k-2}$ subgraphs, and confirming a conjecture about their asymptotic density.

Contribution

It introduces refined edge bounds for $k$-critical graphs involving a measure of disjoint complete subgraphs, extending previous asymptotic density results and confirming a conjecture of Postle.

Findings

Existence of an $oldsymbol{ ext{epsilon}}$-improvement in edge bounds for $k$-critical graphs with $oldsymbol{k ext{≥}33}$.

Linear scaling of disjoint $oldsymbol{K_{k-2}}$ subgraphs with the number of vertices in $k$-Ore graphs.

Validation of Postle's conjecture for $k$-critical $oldsymbol{K_{k-2}}$-free graphs.

Abstract

Recently, Kostochka and Yancey proved that a conjecture of Ore is asymptotically true by showing that every $k$-critical graph satisfies $|E(G)|\geq\left\lceil\left(\frac{k}{2}-\frac{1}{k-1}\right)|V(G)|-\frac{k(k-3)}{2(k-1)}\right\rceil.$ They also characterized the class of graphs that attain this bound and showed that it is equivalent to the set of $k$-Ore graphs. We show that for any $k\geq33$ there exists an $\varepsilon>0$ so that if $G$ is a $k$-critical graph, then $|E(G)|\geq\left(\frac{k}{2}-\frac{1}{k-1}+\varepsilon_k\right)|V(G)|-\frac{k(k-3)}{2(k-1)}-(k-1)\varepsilon T(G)$, where $T(G)$ is a measure of the number of disjoint $K_{k-1}$ and $K_{k-2}$ subgraphs in $G$. This also proves for $k\geq33$ the following conjecture of Postle regarding the asymptotic density: For every $k\geq4$ there exists an $\varepsilon_k>0$ such that if $G$ is a $k$-critical $K_{k-2}$-free graph, then $|E(G)|\geq \left(\frac{k}{2}-\frac{1}{k-1}+\varepsilon_k\right)|V(G)|-\frac{k(k-3)}{2(k-1)}$. As a corollary, our result shows that the number of disjoint $K_{k-2}$ subgraphs in a $k$-Ore graph scales linearly with the number of vertices and, further, that the same is true for graphs whose number of edges is close to Kostochka and Yancey's bound.