Recently, Kostochka and Yancey [7] proved that a conjecture of Ore is asymptotically true by showing that every k-critical graph satisfies $|E(G)|\ge \lceil (\frac{k}{2}-\frac{1}{k-1})|V(G)|-\frac{k(k-3)}{2(k-1)}\rceil$. They also characterized [8] 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 $k\ge 33$ there exists an $\epsilon >0$ so that if G is a k-critical graph, then $|E(G)|\ge (\frac{k}{2}-\frac{1}{k-1}+\epsilon )|V(G)|-\frac{k(k-3)}{2(k-1)}-(k-1)\epsilon T(G)$, 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 $k\ge 33$ the following conjecture of Postle [12] regarding the asymptotic density: For every $k\ge 4$ there exists an ${\epsilon }_k>0$ such that if G is a k-critical $K_{k-2}$-free graph, then $|E(G)|\ge (\frac{k}{2}-\frac{1}{k-1}+{\epsilon }_k)|V(G)|-\frac{k(k-3)}{2(k-1)}$. 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.

最近，Kostochka 和 Yancey [7] 通过证明每个k临界图满足$|乙(G)|\ge \lceil (\frac{ķ}{2}-\frac{1}{ķ-1})|五(G)|-\frac{ķ(ķ-3)}{2(ķ-1)}\rceil$. 他们还描述了 [8] 达到此界限的图类，并表明它等效于k -Ore 图集。我们证明，对于任何$ķ\ge 33$存在一个$\epsilon >0$使得如果G是一个k临界图，那么$|乙(G)|\ge (\frac{ķ}{2}-\frac{1}{ķ-1}+\epsilon )|五(G)|-\frac{ķ(ķ-3)}{2(ķ-1)}-(ķ-1)\epsilon 吨(G)$， 在哪里$吨(G)$是不相交数的量度${ķ}_{ķ-1}$和${ķ}_{ķ-2}$G中的子图。这也证明了$ķ\ge 33$Postle [12] 关于渐近密度的以下猜想： 对于每个$ķ\ge 4$存在一个${\epsilon }_{ķ}>0$使得如果G是k临界的${ķ}_{ķ-2}$-自由图，然后$|乙(G)|\ge (\frac{ķ}{2}-\frac{1}{ķ-1}+{\epsilon }_{ķ})|五(G)|-\frac{ķ(ķ-3)}{2(ķ-1)}$. 作为推论，我们的结果表明，不相交的数量${ķ}_{ķ-2}$k -Ore 图中的子图随顶点数线性缩放，此外，对于边数接近 Kostochka 和 Yancey 界的图也是如此。