Switches are operations which make local changes to the edges of a graph, usually with the aim of preserving the vertex degrees. We study a restricted set of switches, called triangle switches. Each triangle switch creates or deletes at least one triangle. Triangle switches can be used to define Markov chains which generate graphs with a given degree sequence and with many more triangles (3-cycles) than is typical in a uniformly random graph with the same degrees. We show that the set of triangle switches connects the set of all $d$-regular graphs on $n$ vertices, for all $d\geq 3$. Hence, any Markov chain which assigns positive probability to all triangle switches is irreducible on these graphs. We also investigate this question for 2-regular graphs.

翻译：开关是指对图形边缘进行局部变化的操作,通常目的是保护顶点。我们研究了一组限制性开关,称为三角开关。每个三角开关至少创建或删除一个三角。三角开关可用于定义Markov链条,这些链条生成带有一定的度序列的图形,其三角形(3个周期)比同一度统一随机图中常见的多得多。我们显示,三角开关组将所有以美元为单位的以美元为单位的以美元为单位的平方形图与所有以$d\geq 3美元为单位的以美元为单位的平面图相连接。因此,给所有三角开关设定正概率的马尔科夫链条在这些图形上是不可忽略的。我们还为2个普通图形调查了这个问题。