Closing Gaps in Problems related to Hamilton Cycles in Random Graphs and Hypergraphs

We show how to adjust a very nice coupling argument due to McDiarmid in order to prove/reprove in a novel way results concerning Hamilton cycles in various models of random graphs and hypergraphs. In particular, we firstly show that for k > 3, if pnk−1/ log n tends to infinity, then a random k-uniform hypergraph on n vertices, with edge probability p, with high probability (w.h.p.) contains a loose Hamilton cycle, provided that (k − 1)|n. This extends results of Frieze, Dudek and Frieze, and reproves a result of Dudek, Frieze, Loh and Speiss. Secondly, we show that there exists K > 0 such for every p > (K log n)/n the following holds: Let Gn,p be a random graph on n vertices with edge probability p, and suppose that its edges are being colored with n colors uniformly at random. Then, w.h.p the resulting graph contains a Hamilton cycle for which all the colors appear on its edges (a rainbow Hamilton cycle). Bal and Frieze proved the latter statement for graphs on an even number of vertices, where for odd n their p was ω((log n)/n). Lastly, we show that for p = (1 + o(1))(log n)/n, if we randomly color the edge set of a random directed graph Dn,p with (1 + o(1))n colors, then w.h.p. one can find a rainbow Hamilton cycle where all the edges are directed in the same way.