Recently, Kostochka and Yancey proved that a conjecture of Ore is
asymptotically true by showing that every k-critical graph satisfies
β£E(G)β£β₯β(2kββkβ11β)β£V(G)β£β2(kβ1)k(kβ3)ββ.
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
kβ₯33 there exists an Ξ΅>0 so that if G is a k-critical
graph, then
β£E(G)β£β₯(2kββkβ11β+Ξ΅kβ)β£V(G)β£β2(kβ1)k(kβ3)ββ(kβ1)Ξ΅T(G), where T(G) is a measure of the number of disjoint Kkβ1β and
Kkβ2β subgraphs in G. This also proves for kβ₯33 the following
conjecture of Postle regarding the asymptotic density: For every kβ₯4 there
exists an Ξ΅kβ>0 such that if G is a k-critical Kkβ2β-free
graph, then β£E(G)β£β₯(2kββkβ11β+Ξ΅kβ)β£V(G)β£β2(kβ1)k(kβ3)β.
As a corollary, our result shows that the number of disjoint Kkβ2β
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.Comment: 20 page