2010/12/31 by Frieze, Alan, Loh, Po-Shen · 1 citation
#05C38 #05C45 #05C80 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1101.0182
One of the most famous results in the theory of random graphs establishes that the threshold for Hamiltonicity in the Erdos-Renyi random graph Gn,p is around p ~ (log n + log log n) / n. Much research has been done to extend this to increasingly challenging random structures. In particular, a recent result by Frieze determined the asymptotic threshold for a loose Hamilton cycle in the random 3-uniform hypergraph by connecting 3-uniform hypergraphs to edge-colored graphs. In this work, we consider that setting of edge-colored graphs, and prove a result which achieves the best possible first order constant. Specifically, when the edges of Gn,p are randomly colored from a set of (1 + o(1)) n colors, with p = (1 + o(1)) (log n) / n, we show that one can almost always find a Hamilton cycle which has the further property that all edges are distinctly colored (rainbow).