Proceedings 41st Annual Symposium on Foundations of Computer Science (2000)
Redondo Beach, California
Nov. 12, 2000 to Nov. 14, 2000
N. Alon , Dept. of Math., Tel Aviv Univ., Israel
M. Capalbo , Dept. of Math., Tel Aviv Univ., Israel
Y. Kohayakawa , Dept. of Math., Tel Aviv Univ., Israel
V. Rodl , Dept. of Math., Tel Aviv Univ., Israel
A. Rucinski , Dept. of Math., Tel Aviv Univ., Israel
E. Szemeredi , Dept. of Math., Tel Aviv Univ., Israel
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].
graph theory; graph theory; universality; tolerance; positive integers; graphs; vertices; bipartite graphs; maximum degree; sparse universal graphs; random constructions; fault-tolerant bipartite random graph
E. Szemeredi, N. Alon, M. Capalbo, V. Rodl, A. Rucinski and Y. Kohayakawa, "Universality and tolerance," Proceedings 41st Annual Symposium on Foundations of Computer Science(FOCS), Redondo Beach, California, 2000, pp. 14.