Abstract
AbstractA 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 we determine how many random edges one must add to an n‐vertex graph G of minimum degree to ensure that, asymptotically almost surely, the resulting graph contains a perfect Kr‐tiling. As one increases 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, ) and that of Hajnal and Szemerédi [18] (which demonstrates that for the initial graph already houses the desired perfect Kr‐tiling).
Highlights
A significant facet of both extremal graph theory and random graph theory is the study of embeddings
Note that a perfect H-tiling is a generalization of the notion of a perfect matching; perfect matchings correspond to the case when H is a single edge
For every fixed graph H they determined how many random edges one must add to a graph G of linear minimum degree to ensure that a.a.s
Summary
A significant facet of both extremal graph theory and random graph theory is the study of embeddings. In the setting of random graphs, one is interested in the threshold for the property that G(n, p) asymptotically almost surely (a.a.s.) contains a fixed (spanning) subgraph F. A classical line of inquiry in extremal graph theory is to determine the minimum degree threshold that ensures a graph G contains a fixed (spanning) subgraph F. A much studied problem in both the extremal and random settings concerns the case when F is a so-called perfect H-tiling. In this paper we bridge the gap between the random and extremal models for the problem of perfect clique tilings
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