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We first consider the following problem. We are given a fixed perfectmatching $M$ of $[n]$ and we add random edges one at a time until there is aHamilton cycle containing $M$. We show that w.h.p. the hitting time for thisevent is the same as that for the first time there are no isolated vertices inthe graph induced by the random edges. We then use this result for thefollowing problem. We generate random edges and randomly color them black orwhite. A path/cycle is said to \emph{zebraic} if the colors alternate along thepath. We show that w.h.p. the hitting time for a zebraic Hamilton cyclecoincides with every vertex meeting at least one edge of each color. We thenconsider some related problems and extend to multiple colors. We also brieflyconsider directed versions.

Elegantly colored paths and cycles in edge colored random graphs