Tilings in randomly perturbed graphs: Bridging the gap between Hajnal‐Szemerédi and Johansson‐Kahn‐Vu

• Published in: Random structures & algorithms Vol. 58; no. 3; pp. 480 - 516
• Main Authors: Han, Jie, Morris, Patrick, Treglown, Andrew
• Format: Journal Article
• Language: English
• Published: New York John Wiley & Sons, Inc 01.05.2021; Wiley Subscription Services, Inc
• Subjects: Apexes; clique tilings; Graph theory; Graphs; Intervals; random graphs; randomly perturbed structures; Tiling
• ISSN: 1042-9832; 1098-2418
• DOI: 10.1002/rsa.20981

A perfect Kr‐tiling in a graph G is a collection of vertex‐disjoint copies of Kr that together cover all the vertices in G. In this paper we consider perfect Kr‐tilings in the setting of randomly perturbed graphs; a model introduced by Bohman, Frieze, and Martin [7] where one starts with a dense graph and then adds m random edges to it. Specifically, given any fixed 0<α<1−1/r we determine how many random edges one must add to an n‐vertex graph G of minimum degree δ(G)≥αn to ensure that, asymptotically almost surely, the resulting graph contains a perfect Kr‐tiling. As one increases α we demonstrate that the number of random edges required “jumps” at regular intervals, and within these intervals our result is best‐possible. This work therefore closes the gap between the seminal work of Johansson, Kahn and Vu [25] (which resolves the purely random case, that is, α=0) and that of Hajnal and Szemerédi [18] (which demonstrates that for α≥1−1/r the initial graph already houses the desired perfect Kr‐tiling).