三次图的三角形生成过程
Triangle-creation processes on cubic graphs
摘要：
边切换是一种在图中进行局部改变同时保持每个顶点的度的操作。我们引入一个切换动作，称为三角形切换，它创建或删除至少一个三角形。具体地，使移动是选择长度为4的路径$zwvxy$并将其替换为三角形$vxwv$和边$yz$的三角形切换，而使移动执行相反的操作。我们考虑各种马尔可夫链执行随机三角开关，并假设每一个可能的使或打破移动有正概率被执行。我们的第一个结果是，任何这样的马尔可夫链是不可约的所有3-正则图的顶点集$\{1,2,\ldots, n\}$。对于一个特定的，自然的马尔可夫链的这种类型，我们得到了一个非平凡的线性上界和下界的三角形的数量在长期运行。这些界限几乎肯定是在线性时间内得到的，与起始图无关。
英文摘要：
An edge switch is an operation which makes a local change in a graph while maintaining the degree of every vertex. We introduce a switch move, called a triangle switch, which creates or deletes at least one triangle. Specifically, a make move is a triangle switch which chooses a path $zwvxy$ of length 4 and replaces it by a triangle $vxwv$ and an edge $yz$, while a break move performs the reverse operation. We consider various Markov chains which perform random triangle switches, and assume that every possible make or break move has positive probability of being performed. Our first result is that any such Markov chain is irreducible on the set of all 3-regular graphs with vertex set $\{1,2,\ldots, n\}$. For a particular, natural Markov chain of this type, we obtain a non-trivial linear upper and lower bounds on the number of triangles in the long run. These bounds are almost surely obtained in linear time, irrespective of the starting graph.
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