Asymptotically Optimal Tree-Packings in Regular Graphs

Asymptotically Optimal Tree-Packings in Regular Graphs

Let T be a tree with t vertices. Clearly, an n vertex graph contains at most n/t vertex disjoint trees isomorphic to T . In this paper we show that for every > 0, there exists a D( , t) > 0 such that, if d > D( , t) and G is a simple d-regular graph on n vertices, then G contains at least (1− )n/t vertex disjoint trees isomorphic to T .

Asymptoti ally optimal tree-pa kings in regular graphs

Asymptotically Optimal Randomized Tree Embedding in Static Networks Asymptotically Optimal Randomized Tree Embedding in Static Networks

{ The problem of dynamic tree embedding in static networks is studied in this paper. We provide a uniied framework for studying the performance of randomized tree embedding algorithms which allow a newly created tree node to take a random walk of a short distance to reach a processor nearby. In particular, we propose simple randomized algorithms on several most common and important static netwo...

Asymptotically optimal Kk-packings of dense graphs via fractional Kk-decompositions

Let H be a fixed graph. A fractional H-decomposition of a graph G is an assignment of nonnegative real weights to the copies of H in G such that for each e ∈ E(G), the sum of the weights of copies of H containing e is precisely one. An H-packing of a graph G is a set of edge disjoint copies of H in G. The following results are proved. For every fixed k > 2, every graph with n vertices and minim...

Asymptotically optimal induced universal graphs

We prove that the minimum number of vertices of a graph that contains every graph on k vertices as an induced subgraph is (1 + o(1))2(k−1)/2. This improves earlier estimates of Moon, of Bollobás and Thomason and of Alstrup, Kaplan, Thorup and Zwick. The method supplies similarly sharp estimates for the analogous problems for directed graphs, tournaments, bipartite graphs, oriented graphs and mo...