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arXiv 1407.8035math.CO
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关于色数与最小割

On Chromatic Number and Minimum Cut

Meysam Alishahi, Hossein Hajiabolhassan

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摘要：

对于图$G$，树图${\cal T}_{G,t}$具有$G$的所有以$t$为顶点集的树的子图，并且如果两个树的子图是边不相交的，则它们是邻居.此外，$G$的$r^{th}$切割数是将$G$的顶点集划分成两部分的部分之间的边的最小数目，使得每个部分具有至少$r$的大小。证明了如果$t=(1-o(1))n$和$n$足够大，则对于任意一个顶点为$n$的稠密图$G$，树图${\cal T}_{G,t}$的色数等于$G$的$(n-t+1)^{th}$割数.特别地，作为结果，我们证明了，如果$n$是足够大的$G$是一个稠密图，那么生成树图${\cal T}_{G,n}$的色数等于$G$的最小割的大小。证明方法是基于交替图兰数的启发塔克引理，一个等价的组合版本的Borsuk-Ulam定理。

英文摘要：

For a graph $G$, the tree graph ${\cal T}_{G,t}$ has all tree subgraphs of $G$ with $t$ vertices as vertex set and two tree subgraphs are neighbors if they are edge-disjoint. Also, the $r^{th}$ cut number of $G$ is the minimum number of edges between parts of a partition of vertex set of $G$ into two parts such that each part has size at least $r$. We show that if $t=(1-o(1))n$ and $n$ is large enough, then for any dense graph $G$ with $n$ vertices, the chromatic number of the tree graph ${\cal T}_{G,t}$ is equal to the $(n-t+1)^{th}$ cut number of $G$. In particular, as a consequence, we prove that if $n$ is large enough and $G$ is a dense graph, then the chromatic number of the spanning tree graph ${\cal T}_{G,n}$ is equal to the size of the minimum cut of $G$. The proof method is based on alternating Turán number inspired by Tucker's lemma, an equivalent combinatorial version of the Borsuk-Ulam theorem.

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