排除拓扑子式的图的乘积结构
Product structure of graphs excluding a topological minor
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中文总结 AI 辅助
该研究证明了满足树宽条件且排除特定拓扑子式的图的乘积结构，推广了Ding和Oporowski1995年关于有界最大度树宽图的乘积结构的结果。
中文摘要 AI 辅助
我们证明，对于所有正整数h和t，以及每个满足td(X) ≤ h的图X，存在正整数c(X,t)，使得每个树宽tw(G) < t且排除X作为拓扑子式的图G，都同构于某个树宽tw(H) < 2^(h+1)-1的图H与完全图K_{c(X,t)}的笛卡尔积（□）的子图。这推广了Ding和Oporowski（《图论杂志》，1995年）的结果，该结果指出，对于所有正整数Δ和t，存在正整数f(Δ,t)，使得每个树宽tw(G) < t且最大度Δ(G) ≤ Δ的图G，都同构于某个树T与完全图K_{f(Δ,t)}的笛卡尔积的子图。
英文摘要
We prove that, for all positive integers $h$ and $t$ and every graph $X$ with $\mathrm{td}(X) \leq h$, there exists a positive integer $c(X,t)$ such that every graph $G$ with $\mathrm{tw}(G) < t$ that excludes $X$ as a topological minor is isomorphic to a subgraph of $H \boxtimes K_{c(X,t)}$ for some graph $H$ with $\mathrm{tw}(H) < 2^{h+1}-1$. This extends a result by Ding and Oporowski (Journal of Graph Theory; 1995), which states that for all positive integers $Δ$ and $t$, here exists a positive integer $f(Δ,t)$ such that every graph $G$ with $\mathrm{tw}(G)<t$ and $Δ(G)\leqΔ$ is isomorphic to a subgraph of $T \boxtimes K_{f(Δ,t)}$ for some tree $T$.
