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Universality and tolerance

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6 Author(s)
Alon, N. ; Dept. of Math., Tel Aviv Univ., Israel ; Capalbo, M. ; Kohayakawa, Y. ; Rodl, V.
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For any positive integers r and n, let H(r,n) denote the family of graphs on n vertices with maximum degree r, and let H(r,n,n) denote the family of bipartite graphs H on 2n vertices with n vertices in each vertex class, and with maximum degree r. On one hand, we note that any H(r,n)-universal graph must have Ω(n2-2r/) edges. On the other hand, for any n⩾n0(r), we explicitly construct H(r,n)-universal graphs G and Λ on n and 2n vertices, and with O(n2-Ω(1/r log r)) and O(n2-1r/ log1 r/ n) edges, respectively, such that we can efficiently find a copy of any H ε H (r,n) in G 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 log1/2r n is fault-tolerant; for a large enough constant c, even after deleting any α-fraction of the edges of G, the resulting graph is still H(r,α(α)n,α(α)n)-universal for some α: [0,1)→(0,1]

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Foundations of Computer Science, 2000. Proceedings. 41st Annual Symposium on

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