AI 中文总结
研究在稠密无 \(K_t\) 子图的图中 \(k\) - 连通子图的界，通过改进方法，将结构界从 \(O(t\log^3 t)\) 提升到 \(O(t\log^2 t)\)，优化了相关图的阶，从 \(O(t\log^4 t)\) 变为 \(O(t\log^3 t)\)。
AI 中文摘要
Delcourt和Postle将线性哈迪格猜想简化为对具有 \(O(t\log^4 t)\) 个顶点的无 \(K_t\) 子图进行着色。他们证明过程中的一个重要定理表明，每个足够稠密的无 \(K_t\) 子图都包含一个小的、高度连通的子图。本文表明这样的子图可以选得更小。具体而言，存在整数常数 \(C \geq 1\)，对于所有整数 \(t \geq 3\) 和 \(k \geq t\)，每个满足 \(d(G) \geq Ck\) 的无 \(K_t\) 子图 \(G\) 都包含一个非空的 \(k\) - 连通子图 \(H\)，且 \(v(H) \leq C^2 t\log^2 t\)。结构界从 \(O(t\log^3 t)\) 改进到 \(O(t\log^2 t)\)，简化过程中出现的图的阶为 \(O(t\log^3 t)\) 而非 \(O(t\log^4 t)\)。
英文摘要
Delcourt and Postle reduced the Linear Hadwiger Conjecture to coloring $K_t$-minor-free graphs on $O(t\log^4 t)$ vertices. An important theorem in their proof process asserts that every sufficiently dense $K_t$-minor-free graph contains a small, highly connected subgraph. In this paper, we show that such a subgraph can be chosen to be smaller. More precisely, there exists an integer constant $C\geq 1$ such that, for all integers $t\geq 3$ and $k\geq t$, every $K_t$-minor-free graph $G$ with $d(G)\geq Ck$ contains a nonempty $k$-connected subgraph $H$ satisfying $v(H)\leq C^2t\log t$. Thus the structural bound improves from $O(t\log^3 t)$ to $O(t\log t)$. We also give a probabilistic construction showing that the $t\log t$ bound on $v(H)$ is best possible up to a constant factor. Consequently, the graphs occurring in the reduction have order $O(t\log^2 t)$ rather than $O(t\log^4 t)$.
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