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Expanders Are Universal for the Class of All Spanning Trees

Published online by Cambridge University Press:  03 January 2013

DANIEL JOHANNSEN ,
MICHAEL KRIVELEVICH  and
WOJCIECH SAMOTIJ
DANIEL JOHANNSEN
Affiliation:
School of Mathematical Sciences, Tel Aviv University, Tel Aviv, 69978, Israel (e-mail: mail@danieljohannsen.net, krivelev@post.tau.ac.il)
MICHAEL KRIVELEVICH
Affiliation:
School of Mathematical Sciences, Tel Aviv University, Tel Aviv, 69978, Israel (e-mail: mail@danieljohannsen.net, krivelev@post.tau.ac.il)
WOJCIECH SAMOTIJ
Affiliation:
School of Mathematical Sciences, Tel Aviv University, Tel Aviv, 69978, Israel (e-mail: mail@danieljohannsen.net, krivelev@post.tau.ac.il) Trinity College, Cambridge CB2 1TQ, UK (e-mail: samotij@post.tau.ac.il)
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Abstract

A graph is called universal for a given graph class (or, equivalently, -universal) if it contains a copy of every graph in as a subgraph. The construction of sparse universal graphs for various classes has received a considerable amount of attention. There is particular interest in tight -universal graphs, that is, graphs whose number of vertices is equal to the largest number of vertices in a graph from . Arguably, the most studied case is that when is some class of trees. In this work, we are interested in (n,Δ), the class of all n-vertex trees with maximum degree at most Δ. We show that every n-vertex graph satisfying certain natural expansion properties is (n,Δ)-universal. Our methods also apply to the case when Δ is some function of n. Since random graphs are known to be good expanders, our result implies, in particular, that there exists a positive constant c such that the random graph G(n,cn−1/3log2n) is asymptotically almost surely (a.a.s.) universal for (n,O(1)). Moreover, a corresponding result holds for the random regular graph of degree cn2/3log2n. Another interesting consequence is the existence of locally sparse n-vertex (n,Δ)-universal graphs. For example, we show that one can (randomly) construct n-vertex (n,O(1))-universal graphs with clique number at most five. This complements the construction of Bhatt, Chung, Leighton and Rosenberg (1989), whose (n,Δ)-universal graphs with merely O(n) edges contain large cliques of size Ω(Δ). Finally, we show that random graphs are robustly (n,Δ)-universal in the context of the Maker–Breaker tree-universality game.

Keywords

Primary 05C35Secondary 05C0505C8005D40

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Type
Paper
Information
Combinatorics, Probability and Computing , Volume 22 , Issue 2 , March 2013 , pp. 253 - 281
Copyright
Copyright © Cambridge University Press 2013

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