Triangles in randomly perturbed graphs

Abstract

We study the problem of finding pairwise vertex-disjoint triangles in the randomly perturbed graph model, which is the union of any $n$ -vertex graph $G$ satisfying a given minimum degree condition and the binomial random graph $G(n,p)$ . We prove that asymptotically almost surely $G \cup G(n,p)$ contains at least $\min \{\delta (G), \lfloor n/3 \rfloor \}$ pairwise vertex-disjoint triangles, provided $p \ge C \log n/n$ , where $C$ is a large enough constant. This is a perturbed version of an old result of Dirac. Our result is asymptotically optimal and answers a question of Han, Morris, and Treglown [RSA, 2021, no. 3, 480–516] in a strong form. We also prove a stability version of our result, which in the case of pairwise vertex-disjoint triangles extends a result of Han, Morris, and Treglown [RSA, 2021, no. 3, 480–516]. Together with a result of Balogh, Treglown, and Wagner [CPC, 2019, no. 2, 159–176], this fully resolves the existence of triangle factors in randomly perturbed graphs. We believe that the methods introduced in this paper are useful for a variety of related problems: we discuss possible generalisations to clique factors, cycle factors, and $2$ -universality.

The square of a Hamilton cycle in randomly perturbed graphs

“…) significantly strengthens one of our results from [8], where the same result was established for the containment of a triangle factor only.…”

A Ramsey–Turán theory for tilings in graphs

“…This has motivated recent trends in extremal graph theory which have focused on reducing the minimum degree condition necessary by adding an extra condition that provides pseudorandom properties. In this direction, tilings have been explored in popular (strong) notions of pseudorandomness given by so‐called bijumbled graphs and $$$ \left(n,d,\lambda \right) $$$‐graphs [22, 23, 31, 36, 37], as well as in randomly perturbed graphs[5, 9, 10, 24], where one is interested in the amount of random perturbation needed to ensure that a dense graph contains a given tiling. Perhaps the weakest pseudorandom condition one can impose on the host graph is to simply block the existence of large independent sets.…”

“…This was initiated by Bohman et al [6] who studied Hamilton cycles. For tilings, a series of papers [5, 9, 10, 24] have explored the relationship on the minimum degree of a graph $$$ G $$$ and the value of $$$ p $$$ such that $$$ G\cup G\left(n,p\right) $$$ contains a given tiling with high probability (that is, with probability tending to 1 as $$$ n $$$ tends to infinity). Given that the conditions $$$ {\alpha}_r(G)=o(n) $$$, and more pertinently $$$ {\alpha}_r^{\ast }(G)=o(n) $$$, are typical in sparse graphs of a certain density, our results have implications for the randomly perturbed model.…”

Hamiltonicity of graphs perturbed by a random geometric graph

“…Since then, Hamiltonicity has also been considered in randomly perturbed directed graphs [4, 19], hypergraphs [16, 19, 21] and subgraphs of the hypercube [9]. Many other properties have been considered as well (e.g., powers of Hamilton cycles [1, 6, 11, 23], $$F$$‐factors [3, 7, 8, 15], spanning trees [5, 18, 20] or general bounded degree spanning graphs [6]), and in most cases significant improvements on the probability threshold have been achieved. To the best of our knowledge, all of these results consider (hyper/di)graphs perturbed by a binomial random structure, such as $${G}_{n,p}$$, or its $${G}_{n,m}$$ counterpart.…”