arXiv 0812.10642008-12-05math.CO

图的子式与最小度

Graph Minors and Minimum Degree

Gašper Fijavž, David R. Wood

发表信息：: Electronic J. Combinatorics R151, 2010

摘要：

设$\mathcal{D}_k$是所有子图的最小度至多为$k$的图类.然后，$\mathcal{D}_k$因未成年人而关闭。利用Robertson-Seymour图的次定理，我们将$\mathcal{D}_k$刻画为有限族次-极小禁止图，记为$\widehat{\mathcal{D}}_k$ .本文讨论了$\widehat{\mathcal{D}}_k$及其相关主题。我们得到四个主要结果：证明了$(k+1)$ -正则图的顶点数小于${4/3}(k+2)$的图都在$\widehat{\mathcal{D}}_k$中，并且这个界是最佳可能界.我们在$\widehat{\mathcal{D}}_{k+1}$中的图，可以得到一个图在$\widehat{\mathcal{D}}_k$通过添加一个新的顶点。对于$k\leq 3$，$\widehat{\mathcal{D}}_k$中的每个图都是$(k+1)$连通的，但是对于大的$k$，我们给出了连通度为1的$\widehat{\mathcal{D}}_k$中的图.实际上，我们在$\mathcal{D}_k$中构造具有任意块结构的图。我们在$\widehat{\mathcal{D}}_k$中构造了完全多部图，并证明了用连通度、树宽或路宽代替最小度的类似刻画。

英文摘要：

Let $\mathcal{D}_k$ be the class of graphs for which every minor has minimum degree at most $k$. Then $\mathcal{D}_k$ is closed under taking minors. By the Robertson-Seymour graph minor theorem, $\mathcal{D}_k$ is characterised by a finite family of minor-minimal forbidden graphs, which we denote by $\widehat{\mathcal{D}}_k$. This paper discusses $\widehat{\mathcal{D}}_k$ and related topics. We obtain four main results: We prove that every $(k+1)$-regular graph with less than ${4/3}(k+2)$ vertices is in $\widehat{\mathcal{D}}_k$, and this bound is best possible. We characterise the graphs in $\widehat{\mathcal{D}}_{k+1}$ that can be obtained from a graph in $\widehat{\mathcal{D}}_k$ by adding one new vertex. For $k\leq 3$ every graph in $\widehat{\mathcal{D}}_k$ is $(k+1)$-connected, but for large $k$, we exhibit graphs in $\widehat{\mathcal{D}}_k$ with connectivity 1. In fact, we construct graphs in $\mathcal{D}_k$ with arbitrary block structure. We characterise the complete multipartite graphs in $\widehat{\mathcal{D}}_k$, and prove analogous characterisations with minimum degree replaced by connectivity, treewidth, or pathwidth.

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