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arXiv 1703.01477math.CO
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大直径距离一致图

Distance-Uniform Graphs with Large Diameter

Mikhail Lavrov, Po-Shen Loh, Arnau Messegué

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12 pages, 1 figure
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摘要：

一个$ε$ -距离均匀图是这样一个图，其中从每个顶点，除了$ε$ -分数的其余顶点都在某个固定的距离$d$，称为临界距离。研究了具有$n$顶点的$ε$ -距离一致图中$d$的最大可能值.证明了对于$\frac1n \le ε\le \frac1{\log n}$，存在具有临界距离$2^{Ω(\frac{\log n}{\log ε^{-1}})}$的$ε$ -距离一致图，从而反驳了Alon等人关于$d$在$n$中至多是对数的猜想.我们还表明，我们的建设是最好的可能性，在这个意义上，上界的形式$2^{O(\frac{\log n}{\log ε^{-1}})}$ $d$持有所有$ε$和$n$。

英文摘要：

An $ε$-distance-uniform graph is one in which from every vertex, all but an $ε$-fraction of the remaining vertices are at some fixed distance $d$, called the critical distance. We consider the maximum possible value of $d$ in an $ε$-distance-uniform graph with $n$ vertices. We show that for $\frac1n \le ε\le \frac1{\log n}$, there exist $ε$-distance-uniform graphs with critical distance $2^{Ω(\frac{\log n}{\log ε^{-1}})}$, disproving a conjecture of Alon et al. that $d$ can be at most logarithmic in $n$. We also show that our construction is best possible, in the sense that an upper bound on $d$ of the form $2^{O(\frac{\log n}{\log ε^{-1}})}$ holds for all $ε$ and $n$.

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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