5-连通无$\Ke$ minors图的锐密度界
A sharp density bound for 5-connected graphs with no $\Ke$ minor
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中文总结 AI 辅助
本文证明了5-连通图在边数至少4n-9时必含$\Ke$ minor，解决了相关猜想，并加强了根minor定理，该密度界是锐的。
中文摘要 AI 辅助
设$\Ke$是从$K_7$中删除两条独立边得到的图。我们证明每个具有$n\ge7$个顶点且至少有$4n-9$条边的5-连通图包含一个$\Ke$ minor，解决了Dvořák、Norin和Rahman（arXiv预印本2609.17760v1）的猜想1.4。该界是锐的。我们证明了更强的陈述：每个具有$n\ge4$个顶点且至少有$4n-9$条边的4-双轻图要么包含一个$\Ke$ minor，要么包含一个$K_6$子图。在他们的约简框架内，我们加强了根minor定理。我们证明每个根4-密度至少为2的4-轻5-根图具有一个模型，该模型有两个非根顶点且至多有一条与它们关联的缺失边。在临界密度下，约简精确保持密度，这防止它们产生新的$K_6$子图。
英文摘要
Let $\Ke$ be obtained from $K_7$ by deleting two independent edges. We prove that every 5-connected graph on $n\ge7$ vertices with at least $4n-9$ edges contains a $\Ke$ minor, settling Conjecture~1.4 of Dvo\v rák, Norin and Rahman (arXiv preprint 2609.17760v1). The bound is sharp. We prove the stronger statement that every $4$-bilight graph on $n\ge4$ vertices with at least $4n-9$ edges contains either a $\Ke$ minor or a $K_6$ subgraph. Within their reduction framework, we strengthen the rooted-minor theorem. We show that every $4$-light 5-rooted graph of rooted $4$-density at least two has a model with two nonroot vertices and at most one missing edge incident with them. At the critical density, reductions preserve density exactly, which prevents them from creating a new $K_6$ subgraph.
