Universal graphs for bounded-degree trees and planar graphs
SIAM Journal on Discrete Mathematics
Matching and covering the vertices of a random graph by copies of a given graph
Discrete Mathematics
An algorithmic version of the blow-up lemma
Random Structures & Algorithms
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
Regular pairs in sparse random graphs I
Random Structures & Algorithms
An algorithmic Friedman--Pippenger theorem on tree embeddings and applications to routing
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Sparse universal graphs for bounded-degree graphs
Random Structures & Algorithms
Optimal universal graphs with deterministic embedding
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Embedding nearly-spanning bounded degree trees
Combinatorica
Expanders are universal for the class of all spanning trees
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
An improved upper bound on the density of universal random graphs
LATIN'12 Proceedings of the 10th Latin American international conference on Theoretical Informatics
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We prove that asymptotically (as n → ∞) almost all graphs with n vertices and 10d n2--1/2d log 1/d n edges are universal with respect to the family of all graphs with maximum degree bounded by d. Moreover, we provide a polynomial time, deterministic embedding algorithm to find a copy of each bounded degree graph in every graph satisfying some pseudo-random properties. We also prove a counterpart result for random bipartite graphs, where the threshold number of edges is even smaller but the embedding is randomized.