Non-Vertex-Balanced Factors in Random Graphs

TL;DR

This paper proves a conjecture about the threshold for the existence of specific factors in random graphs, extending previous results to more complex graph models and identifying conditions under which the conjecture holds.

Contribution

It confirms the conjectured threshold for H-factors in random graphs for a broad class of graphs and generalizes the main results to multigraphs, digraphs, and multipartite graphs.

Findings

Threshold function correctness for non-vertex-balanced graphs

Extension of results to multigraphs and digraphs

Generalization to multipartite graph models

Abstract

We prove part of a conjecture by Johansson, Kahn and Vu \cite{JKV} regarding threshold functions for the existence of an $H$-factor in a random graph \gnp. We prove that the conjectured threshold function is correct for any graph $H$ which is not covered by its densest subgraphs. We also demonstrate that the main result of \cite{JKV} generalises to multigraphs, digraphs, and a multipartite model.