Seminar
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| Location: | SLMath: Eisenbud Auditorium, Online/Virtual |
Clique factors in randomly augmented graphs
Randomly augmented graphs interpolate between extremal problems and random graph thresholds. Specifically, a randomly augmented graph G^p = G_a U G(n,p) is obtained by taking a deterministic n-vertex graph G_a with minimum degree an and adding the edges of the binomial random graphG(n,p) defined on the same vertex set. For which value p does the graph G^p contain a K_r-factor with high probability?
We will survey some results and proof ideas in this area which naturally combines extremal probabilistic techniques. Specifically, the order of magnitude of the minimal value of p has been determined for 'most' values of the minimum degree a (see Han, Morris, and Treglown [RSA, 2021] and Balogh, Treglown, and Wagner [CPC, 2019]). However, we find that the threshold density exhibits surprising polynomial jumps for values a= 1-s/r (where s is an integer) compared to the surrounding intervals.
Joint work with Sylwia Antoniuk and Christian Reiher.
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