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arXiv 1809.02582math.CO
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直径固定的有向图的卵石化

Pebbling on Directed Graphs with Fixed Diameter

John Asplund, Franklin Kenter

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15 pages, 3 figures
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摘要：

卵石是一种在图上玩的游戏。单个玩家被给予图和卵石的配置，并且可以通过从一个顶点移除2个卵石并将一个卵石放置在相邻顶点以最终使一个卵石到达预定顶点来进行卵石移动。卵石数量$π(G)$是卵石的最小数量，无论其确切配置如何，玩家都可以使用卵石移动使卵石到达任何预定的顶点。先前的工作已经将$π(G)$与$G$的直径相关联。Clarke、Hochberg和Hurlbert证明了$n$顶点上直径为2的每个连通无向图都有$π(G) = n$，除非它属于一个特殊的图族，由那些可以以特定方式构造的图组成;在这种情况下，$π(G) = n +1$。通过推广Chan和Godbole的一个结果，Postle证明了对于直径为$d$的图，$π(G) \le n 2^{\lceil \frac{d}{2} \rceil} (1+o_n(1))$ .在这篇文章中，我们继续这项研究有关卵石和直径的重点是有向图。这导致了一些令人惊讶的结果。首先，我们证明了在有向有向图$G$中（在这个意义上，如果$i \to j$，那么我们就不能有$j \to i$），如果$G$的直径为2，则确实是$π(G) = n$或$n + 1$，如果$π(G) = n+1$，则有向图具有非常特殊的结构。在直径为2的一般有向图的情况下（即，如果$i \to j$，我们可能有也可能没有弧$j \to i$），我们证明了$π(G)$可以和$\frac32 n + 1$一样大，而且这个界是尖锐的。更一般地说，我们表明，一般有向图，$π(G) \le 2^d n / d + f(d)$其中$f(d)$是一些功能，只有$d$。

英文摘要：

Pebbling is a game played on a graph. The single player is given a graph and a configuration of pebbles and may make pebbling moves by removing 2 pebbles from one vertex and placing one at an adjacent vertex to eventually have one pebble reach a predetermined vertex. The pebbling number, $π(G)$, is the minimum number of pebbles such that regardless of their exact configuration, the player can use pebbling moves to have a pebble reach any predetermined vertex. Previous work has related $π(G)$ to the diameter of $G$. Clarke, Hochberg, and Hurlbert demonstrated that every connected undirected graph on $n$ vertices with diameter 2 has $π(G) = n$ unless it belongs to an exceptional family of graphs, consisting of those that can be constructed in a specific manner; in which case $π(G) = n +1$. By generalizing a result of Chan and Godbole, Postle showed that for a graph with diameter $d$, $π(G) \le n 2^{\lceil \frac{d}{2} \rceil} (1+o_n(1))$. In this article, we continue this study relating pebbling and diameter with a focus on directed graphs. This leads to some surprising results. First, we show that in an oriented directed graph $G$ (in the sense that if $i \to j$ then we cannot have $j \to i$), it is indeed the case that if $G$ has diameter 2, $π(G) = n$ or $n + 1$, and if $π(G) = n+1$, the directed graph has a very particular structure. In the case of general directed graphs (that is, if $i \to j$, we may or may not have an arc $j \to i$) with diameter 2, we show that $π(G)$ can be as large as $\frac32 n + 1$, and further, this bound is sharp. More generally, we show that for general directed graphs, $π(G) \le 2^d n / d + f(d)$ where $f(d)$ is some function of only $d$.

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