A randomized construction of high girth regular graphs

We describe a new random greedy algorithm for generating regular graphs of high girth: Let k ≥ 3 and c ∈ (0, 1) be fixed. Let (Formula presented.) ℕ be even and set (Formula presented.). Begin with a Hamilton cycle G on n vertices. As long as the smallest degree (Formula presented.), choose, uniformly at random, two vertices u, v ∈ V(G) of degree (Formula presented.) 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 (Formula presented.) labeled k-regular n-vertex graphs with girth at least g.