Tiling randomly perturbed bipartite graphs

TL;DR

This paper determines the precise threshold for perfect tilings of bipartite graphs with complete bipartite subgraphs in the context of randomly perturbed graphs with linear minimum degree.

Contribution

It establishes the exact conditions under which a perfect $K_{h,h}$-tiling exists in randomly perturbed bipartite graphs, extending previous work on tiling thresholds.

Findings

Identifies the threshold for perfect $K_{h,h}$-tilings in perturbed bipartite graphs.

Extends tiling threshold results to the setting of random perturbations.

Provides a framework for understanding tilings in graphs with combined deterministic and random structures.

Abstract

A perfect $H$-tiling in a graph $G$ is a collection of vertex-disjoint copies of a graph $H$ in $G$ that covers all vertices of $G$. Motivated by papers of Bush and Zhao and of Balogh, Treglown, and Wagner, we determine the threshold for the existence of a perfect $K_{h,h}$-tiling of a randomly perturbed bipartite graph with linear minimum degree.