No abstract

The main concrete result of this paper is the first explicit construction of constant degree lossless expanders. In these graphs, the expansion factor is almost as large as possible:(1 − )D, where D is the degree and is an arbitrarily small constant. The best previous explicit constructions gave expansion factor D/2, which is too weak for many applications. The D/2 bound was obtained via the eigenvalue method, and is known that that method cannot give better bounds.The main abstract contribution of this paper is the introduction and initial study of randomness conductors, a notion which generalizes extractors, expanders, condensers and other similar objects. In all these functions, certain guarantee on the input "entropy" is converted to a guarantee on the output "entropy". For historical reasons, specific objects used specific guarantees of different flavors. We show that the flexibility afforded by the conductor definition leads to interesting combinations of these objects, and to better constructions such as those above. * A full version of this paper will be posted on the Electronic Colloquium on Computational Complexity,

ABSTRACT:Let H be a family of graphs. A graph T is H-universal if it contains a copy of each H ∈ H as a subgraph. Let H(k, n) denote the family of graphs on n vertices with maximum degree at most k. For all positive integers k and n, we construct an H(k, n)-universal graph T with O k (n 2− 2 k log 4 k n) edges and exactly n vertices. The number of edges is almost as small as possible, as (n 2−2/k ) is a lower bound for the number of edges in any such graph. The construction of T is explicit, whereas the proof of universality is probabilistic and is based on a novel graph decomposition result and on the properties of random walks on expanders.

For any positive integers r and n, let Z ( r , n) denote the family of graphs on n vertices with maximum degree r, and let %(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 Z ( r , n)-universal graph must have fl(n2-2/r) edges. On the other hand, for any n 2 no (r), we explicitly construct X(r,n)-universal graphs G and A on n and 2n vertices, and with 0(n2-'(*)) and O ( n 2 -f log1/' n ) edges, respectively, such that we can eficientlyfind a copy of any H E X ( 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-k log'/2i n is fault-tolerant; for a large enough constant c, even afer deleting any a-fraction of the edges of G, the resulting graph is still N ( r , a ( a ) n , a(a)n)-universal for some a : [O, 1) + (0,1]. r 1 2. This lower bound follows from the obvious inequality &rn/2 (7) 2 131(r, n)l and the well known (see, e.g., [ 111, Cor. 9.8, page 239) asymptotic formula for the number Lr,, of all labelled r-regular graphs on n vertices, n r even: Let M ( r , n) = M be the minimum number of edges in an N(r, n)-universal graph. The inequalities (7) 5 ( y ) i for i 5 r n / 2 and 131(r,n)l 2 L,,,/n! yield the claimed lower bound M ( r , n ) = fl(n2-2/r) 14 0-7695-0850-2/00 $10.00 0 2000 IEEE S . Janson, T. tuczak and A. Ruciriski, Random Graphs, Wiley, New York, 2000. Y. Kohayakawa and V. Rodl, Regular pairs in sparse random graphs I, submitted, 2000.Y. Kohayakawa, V. Rodl and E. SzemerBdi, in preparation.