论文标题
在燃烧芯片的游戏长度上
On lengths of burn-off chip-firing games
论文作者
论文摘要
我们继续研究[离散数学的燃烧芯片游戏。理论。计算。科学。 15（2013），没有。 1，121-132; MR3040546]和[澳大利亚。 J. Combin。 68（2017），没有。 3，330-345; MR3656659]。后一篇文章通过从图$ g $的顶点集中选择连续的种子来介绍随机性。游戏的长度是随着激动的芯片配置传递到轻松的状态，将射击的顶点（通过向每个邻居发送芯片并歼灭一个芯片）的顶点数量。本文确定了长期燃烧游戏中游戏长度的概率分布。我们的主要结果给出了对$（c，v）$的确切计数，带有$ c $ a宽松的法律配置和$ v $ a种子，对应于每个可能的长度。为了支持，我们提供了自己的证明，证明了$ g $上放松的法律配置的集合$ \ Mathcal {r} $的众所周知的等法性，以及$ g $的圆锥形$ g^*$中的一组跨树。我们提出了该对应关系的算法，公约的证明。
We continue our studies of burn-off chip-firing games from [Discrete Math. Theor. Comput. Sci. 15 (2013), no. 1, 121-132; MR3040546] and [Australas. J. Combin. 68 (2017), no. 3, 330-345; MR3656659]. The latter article introduced randomness by choosing successive seeds uniformly from the vertex set of a graph $G$. The length of a game is the number of vertices that fire (by sending a chip to each neighbor and annihilating one chip) as an excited chip configuration passes to a relaxed state. This article determines the probability distribution of the game length in a long sequence of burn-off games. Our main results give exact counts for the number of pairs $(C,v)$, with $C$ a relaxed legal configuration and $v$ a seed, corresponding to each possible length. In support, we give our own proof of the well-known equicardinality of the set $\mathcal{R}$ of relaxed legal configurations on $G$ and the set of spanning trees in the cone $G^*$ of $G$. We present an algorithmic, bijective proof of this correspondence.
