Abstract
We solve a problem of Krivelevich, Kwan and Sudakov concerning the threshold for the containment of all bounded degree spanning trees in the model of randomly perturbed dense graphs. More precisely, we show that, if we start with a dense graph Gα on n vertices with δ(Gα) ≥ αn for α > 0 and we add to it the binomial random graph G(n,C/n), then with high probability the graph Gα∪G(n,C/n) contains copies of all spanning trees with maximum degree at most Δ simultaneously, where C depends only on α and Δ.
Highlights
Many problems from extremal graph theory concern Dirac-type questions. These ask for asymptotically optimal conditions on the minimum degree δpGnq for an n-vertex graph Gn to contain a given spanning graph Fn
A large branch of the theory of random graphs studies when random graphs typically contain a copy of random graph, where each a given of thesnp2 ̆anpnoinssgibslteruecdtguerse
We find a subtree T1 of T of small linear size, say βn with β ! α, and we embed this subtree T1 into H using a randomized algorithm
Summary
Many problems from extremal graph theory concern Dirac-type questions. These ask for asymptotically optimal conditions on the minimum degree δpGnq for an n-vertex graph Gn to contain a given spanning graph Fn. Typically, there exists a constant α ą 0 (depending on the family pFiqiě1) such that δpGnq ě αn implies Fn Ď Gn. A prime example is Dirac’s theorem [9] stating that δpGnq ě n{2 ensures that Gn is Hamiltonian if n ě 3. Since containing a copy of (a sequence of graphs) Fn is a monotone property, there exists a threshold function p “ ppnq : N Ñ r0, 1s such that, if p “ oppq, limnÑ8 PrFn Ď Gpn, pqs “ 0, whereas, if p “ ωppq, limnÑ8 PrFn Ď Gpn, pqs “ 1. A famous result of Koršunov [15] and Pósa [22] asserts that the threshold for Hamiltonicity in Gpn, pq is plog nq{n
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