On the Last New Vertex Visited by a Random Walk in a Directed Graph

IF 1 Q1 MATHEMATICS
Calum Buchanan, P. Horn, Puck Rombach
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引用次数: 0

Abstract

Consider a simple graph in which a random walk begins at a given vertex. It moves at each step with equal probability to any neighbor of its current vertex, and ends when it has visited every vertex. We call such a random walk a random cover tour. It is well known that cycles and complete graphs have the property that a random cover tour starting at any vertex is equally likely to end at any other vertex. Ronald Graham asked whether there are any other graphs with this property. In 1993, L\'aszlo Lov\'asz and Peter Winkler showed that cycles and complete graphs are the only undirected graphs with this property. We strengthen this result by showing that cycles and complete graphs (with all edges considered bidirected) are the only directed graphs with this property.
关于有向图随机漫步最后访问的新顶点
考虑一个简单的图,其中随机游走从给定的顶点开始。在每一步中,它以等概率移动到其当前顶点的任何邻居,并在访问每个顶点时结束。我们称这种随机漫步为随机封面之旅。众所周知,循环和完全图具有这样的性质,即从任何顶点开始的随机覆盖巡回都同样有可能在任何其他顶点结束。Ronald Graham问是否还有其他的图具有这个性质。1993年,L\ aszlo Lov\ asz和Peter Winkler证明了循环和完全图是唯一具有这种性质的无向图。我们通过证明循环和完全图(所有边都被认为是双向的)是唯一具有这个性质的有向图来加强这个结果。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Discrete Mathematics Letters
Discrete Mathematics Letters Mathematics-Discrete Mathematics and Combinatorics
CiteScore
1.50
自引率
12.50%
发文量
47
审稿时长
12 weeks
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