arXiv 1601.011292016-01-06math.CO

极小正规图覆盖

Minimal normal graph covers

David Gajser, Bojan Mohar

备注：: 16 pages, 6 figures

摘要：

一个图是正规的，如果它允许一个团覆盖$\mathcal C$和一个稳定集覆盖$\mathcal S$，使得$\mathcal C$中的每个团和$\mathcal S$中的每个稳定集有一个公共顶点。对$(\mathcal{C,S})$是图的正规覆盖。我们给出了正规覆盖的极值性质。对于正整数$c,s$，如果一个顶点数为$n$的图的正规覆盖的团的大小至多为$c$，稳定集的大小至多为$s$，则$c+s\geq\log_2(n)$ .对于无穷多个$n$，我们也给出了一个图的结构与$n$顶点，允许一个正常的覆盖团和稳定集的大小小于$0.87\log_2(n)$。进一步证明了对所有的$n$，存在一个正规图，它的顶点数为$n$，团数为$Θ(\log_2(n))$，独立数为$Θ(\log_2(n))$ .当$c$或$s$非常小时，我们可以描述所有具有尽可能多的顶点数的正规图，这些顶点数允许正规覆盖的团的大小最多为$c$，稳定集的大小最多为$s$。然而，即使对于$c$和$s$的适度小的值，这样的极值图仍然难以捉摸。

英文摘要：

A graph is normal if it admits a clique cover $\mathcal C$ and a stable set cover $\mathcal S$ such that each clique in $\mathcal C$ and each stable set in $\mathcal S$ have a vertex in common. The pair $(\mathcal{C,S})$ is a normal cover of the graph. We present the following extremal property of normal covers. For positive integers $c,s$, if a graph with $n$ vertices admits a normal cover with cliques of sizes at most $c$ and stable sets of sizes at most $s$, then $c+s\geq\log_2(n)$. For infinitely many $n$, we also give a construction of a graph with $n$ vertices that admits a normal cover with cliques and stable sets of sizes less than $0.87\log_2(n)$. Furthermore, we show that for all $n$, there exists a normal graph with $n$ vertices, clique number $Θ(\log_2(n))$ and independence number $Θ(\log_2(n))$. When $c$ or $s$ are very small, we can describe all normal graphs with the largest possible number of vertices that allow a normal cover with cliques of sizes at most $c$ and stable sets of sizes at most $s$. However, such extremal graphs remain elusive even for moderately small values of $c$ and $s$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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