Pattern Colored Hamilton Cycles in Random Graphs

We consider the existence of patterned Hamilton cycles in randomly colored random graphs. Given a string $\mathrm{\Pi }$ over a set of colors $\{1,2,\dots ,r\}$, we say that a Hamilton cycle is $\mathrm{\Pi }$-colored if the pattern repeats at intervals of length $\mathrm{\mid }\mathrm{\Pi }\mathrm{\mid }$ 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 $\mathrm{\Pi }$-connected.