Spanning trees in randomly perturbed graphs
Abstract
A classical result of Komlós, Sárközy and Szemerédi states that every $n$vertex graph with minimum degree at least $(1/2+ o(1))n$ contains every $n$vertex tree with maximum degree $O(n/\log{n})$ as a subgraph, and the bounds on the degree conditions are sharp. On the other hand, Krivelevich, Kwan and Sudakov recently proved that for every $n$vertex graph $G_\alpha$ with minimum degree at least $\alpha n$ for any fixed $\alpha >0$ and every $n$vertex tree $T$ with bounded maximum degree, one can still find a copy of $T$ in $G_\alpha$ with high probability after adding $O(n)$ randomlychosen edges to $G_\alpha$. We extend their results to trees with unbounded maximum degree. More precisely, for a given $n^{o(1)}\leq \Delta\leq cn/\log n$ and $\alpha>0$, we determine the precise number (up to a constant factor) of random edges that we need to add to an arbitrary $n$vertex graph $G_\alpha$ with minimum degree $\alpha n$ in order to guarantee a copy of any fixed $n$vertex tree $T$ with maximum degree at most~$\Delta$ with high probability.
 Publication:

arXiv eprints
 Pub Date:
 March 2018
 DOI:
 10.48550/arXiv.1803.04958
 arXiv:
 arXiv:1803.04958
 Bibcode:
 2018arXiv180304958J
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 41 pages