Spanning $F$-cycles in random graphs

TL;DR

This paper extends threshold results for spanning structures in random graphs, providing new conditions for various cycles and answering open questions about overlapping cycles and specific spanning configurations.

Contribution

It introduces a general sufficient condition for determining thresholds of spanning structures and applies it to new classes of cycles, including overlapping $C_4$ and $K_r$-cycles.

Findings

Threshold for spanning $C_4$ with overlapping edges determined

Threshold for spanning $K_r$-cycles with edge overlaps established

General sufficient condition for thresholds of spanning graphs introduced

Abstract

We extend a recent argument of Kahn, Narayanan and Park (Proceedings of the AMS, to appear) about the threshold for the appearance of the square of a Hamilton cycle to other spanning structures. In particular, for any spanning graph, we give a sufficient condition under which we may determine its threshold. As an application, we find the threshold for a set of cyclically ordered copies of $C_4$ that span the entire vertex set, so that any two consecutive copies overlap in exactly one edge and all overlapping edges are disjoint. This answers a question of Frieze. We also determine the threshold for edge-overlapping spanning $K_r$-cycles.