Pattern Colored Hamilton Cycles in Random Graphs

TL;DR

This paper investigates the emergence of patterned Hamilton cycles in randomly colored graphs, establishing the precise moment they appear during the graph's evolution.

Contribution

It introduces the concept of patterned Hamilton cycles in randomly colored graphs and determines the exact hitting time for their existence.

Findings

Identifies the threshold for the appearance of patterned Hamilton cycles.

Establishes the hitting time for onnectedness related to patterns.

Provides probabilistic bounds for the emergence of these cycles.

Abstract

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.