We describe a new random greedy algorithm for generating regular graphs of high girth: Let k ≥ 3 and c ∈ (0, 1) be fixed. Let ℕ be even and set . Begin with a Hamilton cycle G on n vertices. As long as the smallest degree , choose, uniformly at random, two vertices u, v ∈ V(G) of degree whose distance is at least g − 1. If there are no such vertex pairs, abort. Otherwise, add the edge uv to E(G). We show that with high probability this algorithm yields a k‐regular graph with girth at least g. Our analysis also implies that there are labeled k‐regular n‐vertex graphs with girth at least g.

中文翻译：

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高周长正则图的随机构造

我们描述了用于产生高周长正则图一个新的随机贪婪算法：设ķ  ≥3和Ç  ∈（0，1）是固定的。让ℕ均匀。从n个顶点上的哈密顿循环G开始。只要最小的程度，选择，均匀地随机的，两个顶点ü，  v  ∈  V（G ^度）的距离为至少克 - 1。如果没有这样的顶点对，中止。否则，将边缘uv添加到E（G）。我们证明该算法很有可能产生一个k周长至少为g的正则图。我们的分析还暗示，有标记的k正则n顶点图，其周长至少为g。