Abstract
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.
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