Universality and tolerance

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 /spl Omega/(n/sup 2-2/r/) edges. On the other hand, for any n/spl ges/n/sub 0/(r), we explicitly construct H(r,n)-universal graphs G and /spl Lambda/ on n and 2n vertices, and with O(n/sup 2-/spl Omega//(1/r log r)) and O(n/sup 2-1/r/ log/sup 1/r/ n) edges, respectively, such that we can efficiently find a copy of any H /spl epsiv/ 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/sup -1/2r/ log/sup 1/2r/ n is fault-tolerant; for a large enough constant c, even after deleting any /spl alpha/-fraction of the edges of G, the resulting graph is still H(r,/spl alpha/(/spl alpha/)n,/spl alpha/(/spl alpha/)n)-universal for some /spl alpha/: [0,1)/spl rarr/(0,1].