arXiv 1306.18032013-06-07math.CO

给定最大度的图中完全子图的最大个数

The maximum number of complete subgraphs in a graph with given maximum degree

Jonathan Cutler, A. J. Radcliffe

备注：: 14 pages

摘要：

涉及图的子结构计数的极值问题在图论中有着悠久的历史。例如，根据Kahn-Zhao定理，$d$ -正则图中$n$顶点上的独立集的个数至多为$(2^{d+1}-1)^{n/2d}$。Galvin将正则性约束放宽到最小度条件，证明了对于$n\geq 2d$，具有$δ(G)\geq d$的图中的独立集的数目至多是$K_{d,n-d}$中的独立集的数目。本文给出了$d$ -正则图中独立集个数的一个下界，它反映了Kahn-Zhao定理中的上界.本文的主要结果是Galvin猜想的一个加强形式的证明，也包括$n\leq 2d$的情形。我们发现从$\complement{G}$的角度来解决这个问题是很方便的。换句话说，我们给出了一个图$G$在$n$顶点上的完全子图个数的上界，该上界对$Δ(G)\leq r$和$n$都有效。

英文摘要：

Extremal problems involving the enumeration of graph substructures have a long history in graph theory. For example, the number of independent sets in a $d$-regular graph on $n$ vertices is at most $(2^{d+1}-1)^{n/2d}$ by the Kahn-Zhao theorem. Relaxing the regularity constraint to a minimum degree condition, Galvin conjectured that, for $n\geq 2d$, the number of independent sets in a graph with $δ(G)\geq d$ is at most that in $K_{d,n-d}$. In this paper, we give a lower bound on the number of independent sets in a $d$-regular graph mirroring the upper bound in the Kahn-Zhao theorem. The main result of this paper is a proof of a strengthened form of Galvin's conjecture, covering the case $n\leq 2d$ as well. We find it convenient to address this problem from the perspective of $\complement{G}$. In other words, we give an upper bound on the number of complete subgraphs of a graph $G$ on $n$ vertices with $Δ(G)\leq r$, valid for all values of $n$ and $r$.

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。

摘要：

英文摘要：

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