A ug 2 00 5 RAINBOW HAMILTON CYCLES IN RANDOM REGULAR GRAPHS SVANTE JANSON

Rainbow Hamilton cycles in random regular graphs

A rainbow subgraph of an edge-coloured graph has all edges of distinct colours. A random d-regular graph with d even, and having edges coloured randomly with d/2 of each of n colours, has a rainbow Hamilton cycle with probability tending to 1 as n→∞, for fixed d ≥ 8.

A ug 2 00 5 RAINBOW HAMILTON CYCLES IN RANDOM REGULAR GRAPHS

A rainbow subgraph of an edge-coloured graph has all edges of distinct colours. A random d-regular graph with d even, and having edges coloured randomly with d/2 of each of n colours, has a rainbow Hamilton cycle with probability tending to 1 as n → ∞, provided d ≥ 8.

Hamilton Cycles in a Random Tournament

The number of Hamilton cycles in a random tournament is asymptotically normally distributed.

The Numbers of Spanning Trees, Hamilton cycles and Perfect Matchings in a Random Graph

. C O ] 8 D ec 2 00 8 ON VERTEX , EDGE , AND VERTEX - EDGE RANDOM GRAPHS

We consider three classes of random graphs: edge random graphs, vertex random graphs, and vertex-edge random graphs. Edge random graphs are Erd˝ os-Rényi random graphs [5, 6], vertex random graphs are generalizations of geometric random graphs [16], and vertex-edge random graphs generalize both. The names of these three types of random graphs describe where the randomness in the models lies: in...