UNIVERSALITY AND TOLERANCE (Extended Abstract)

For any positive integersr and n, let H(r, n) denote the family of graphs onn vertices with maximum degree r, and letH(r, n, n) denote the family of bipartite graphs H on 2n vertices withn vertices in each vertex class, and with maximum degreer. On one hand, we note that any H(r, n)-universal graph must have Ω(n2−2/r) edges. On the other hand, for anyn ≥ n0(r), we explicitly construct H(r, n)-universal graphsG and Λ on n and 2n vertices, and withO(n2−Ω( 1 r log r ) andO(n2− 1 r log n) edges, respectively, such that we can efficiently find a copy of any H ∈ H(r, n) inG deterministically. We also achieve sparse universal graphs using random constructions. Finally, we show that the bipartite random graph G = G(n, n, p), with p = cn− 1 2r log n is fault-tolerant; for a large enough constantc, even after deleting any α-fraction of the edges of G, the resulting graph is still H(r, a(α)n, a(α)n)-universal for somea : [0, 1)→ (0, 1]. ∗Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel. Research supported in part by a USA-Israeli BSF grant, by the Israel Science Foundation and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University. Email: noga@math.tau.ac.il. †Department of Mathematical Sciences, The Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD. Supported by NSF grant CCR98210-58 and ARO grant DAAH04-96-1-0013. Email: capalbo@mts.jhu.edu. ‡Instituto de Mateḿatica e Estat ı́stica, Universidade de S ão Paulo, Brazil. Research partially supported by FAPESP (Proc. 96/04505–2), by CNPq (Proc. 300334/93–1), and by MCT/FINEP under PRONEX project 107/97. Email: yoshi@ime.usp.br §Department of Mathematics and Computer Science, Emory University, Atlanta. Research supported by NSF grant DMS 9704114. Email: rodl@mathcs.emory.edu ¶Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Pozná n, Poland. Research supported by KBN grant 2 P03A 032 16. Part of this research was done during the author’s visit to Emory University. Email: rucinski@amu.edu.pl ‖Department of Computer Science, Rutgers University. Email: szemered@cs.rutgers.edu