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Embedding spanning trees in random graphs
 SIAM Journal on Discrete Mathematics
"... We prove that if T is a tree on n vertices with maximum degree ∆ and the edge probability p(n) satisfies: np ≥ C max{ ∆ log n, n ɛ} for some constant ɛ> 0, then with high probability the random graph G(n, p) contains a copy of T. The obtained bound on the edge probability is shown to be essential ..."
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We prove that if T is a tree on n vertices with maximum degree ∆ and the edge probability p(n) satisfies: np ≥ C max{ ∆ log n, n ɛ} for some constant ɛ> 0, then with high probability the random graph G(n, p) contains a copy of T. The obtained bound on the edge probability is shown to be essentially tight for ∆ = n Θ(1). 1
Extremal graph packing problems: Oretype versus Diractype
"... We discuss recent progress and unsolved problems concerning extremal graph packing, emphasizing connections between Diractype and Oretype problems. Extra attention is paid to coloring, and especially equitable coloring, of graphs. 1 ..."
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We discuss recent progress and unsolved problems concerning extremal graph packing, emphasizing connections between Diractype and Oretype problems. Extra attention is paid to coloring, and especially equitable coloring, of graphs. 1
Expanders Are Universal for the Class of All Spanning Trees
, 2012
"... Given a class of graphs F, we say that a graph G is universal for F, or Funiversal, if every H ∈ F is contained in G as a subgraph. The construction of sparse universal graphs for various families F has received a considerable amount of attention. One is particularly interested in tight Funiversal ..."
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Given a class of graphs F, we say that a graph G is universal for F, or Funiversal, if every H ∈ F is contained in G as a subgraph. The construction of sparse universal graphs for various families F has received a considerable amount of attention. One is particularly interested in tight Funiversal graphs, i. e., graphs whose number of vertices is equal to the largest number of vertices in a graph from F. Arguably, the most studied case is that when F is some class of trees. Given integers n and ∆, we denote by T (n, ∆) the class of all nvertex trees with maximum degree at most ∆. In this work, we show that every nvertex graph satisfying certain natural expansion properties is T (n, ∆)universal or, in other words, contains every spanning tree of maximum degree at most ∆. Our methods also apply to the case when ∆ is some function of n. The result has a few very interesting implications. Most importantly, since random graphs are known to be good expanders, we obtain that the random graph G(n, p) is asymptotically almost surely (a.a.s.) universal for the class of all bounded
Spanning embeddings of arrangeable graphs with sublinear bandwidth
 In preparation
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ON THE TREE PACKING CONJECTURE
"... The Gyárfás tree packing conjecture states that any set of n−1 trees T1, T2,..., Tn−1 such that Ti has n − i + 1 vertices pack into Kn (for n large enough). We show that t = 1 10 n1/4 trees T1, T2,..., Tt such that Ti has n − i + 1 vertices pack into Kn+1 (for n large enough). We also prove that an ..."
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The Gyárfás tree packing conjecture states that any set of n−1 trees T1, T2,..., Tn−1 such that Ti has n − i + 1 vertices pack into Kn (for n large enough). We show that t = 1 10 n1/4 trees T1, T2,..., Tt such that Ti has n − i + 1 vertices pack into Kn+1 (for n large enough). We also prove that any set of t = 1 10 n1/4 trees T1, T2,..., Tt such that no tree is a star and Ti has n − i + 1 vertices pack into Kn (for n large enough). Finally, we prove that t = 1 4 n1/3 trees T1, T2,..., Tt such that Ti has n − i + 1 vertices pack into Kn as long as each tree has maximum degree at least 2n2/3 (for n large enough). One of the main tools used in the paper is the famous spanning tree embedding theorem of Komlós, Sárközy and Szemerédi [15].
Bandwidth, treewidth, separators, expansion, and universality
"... We prove that planar graphs with bounded maximum degree have sublinear bandwidth. As a consequence for each γ> 0 every nvertex graph with minimum degree (3 4 + γ)n contains a copy of every boundeddegree planar graph on n vertices. The proof relies on the fact that planar graphs have small sepa ..."
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We prove that planar graphs with bounded maximum degree have sublinear bandwidth. As a consequence for each γ> 0 every nvertex graph with minimum degree (3 4 + γ)n contains a copy of every boundeddegree planar graph on n vertices. The proof relies on the fact that planar graphs have small separators. Indeed, we show more generally that for any class of boundeddegree graphs the concepts of sublinear bandwidth, sublinear treewidth, the absence of big expanders as subgraphs, and the existence of small separators are equivalent.