Pattern Colored Hamilton Cycles in Random Graphs

We consider the existence of patterned Hamilton cycles in randomly colored random graphs. Given a string $\Pi$ over a set of colors $\{1,2,\ldots,r\}$, we say that a Hamilton cycle is $\Pi$-colored if the pattern repeats at intervals of length $|\Pi|$ as we go around the cycle. We prove a hitting time for the existence of such a cycle. We also prove a hitting time result for a related notion of $\Pi$-connected.