A Distributed algorithm to find Hamiltonian cycles in Gnp random graphs

In this paper, we present a distributed algorithm to find Hamiltonian cycles in random binomial graphs Gnp. The algorithm works on a synchronous distributed setting by first creating a small cycle, then covering almost all vertices in the graph with several disjoint paths, and finally patching these paths and the uncovered vertices to the cycle. Our analysis shows that, with high probability, our algorithm is able to find a Hamiltonian cycle in Gnp when p_n=omega(sqrt{log n}/n^{1/4}). Moreover, we conduct an average case complexity analysis that shows that our algorithm terminates in expected sub-linear time, namely in O(n^{3/4+epsilon}) pulses.Postprint (published version

Approximately Counting Embeddings into Random Graphs

Let H be a graph, and let C_H(G) be the number of (subgraph isomorphic) copies of H contained in a graph G. We investigate the fundamental problem of estimating C_H(G). Previous results cover only a few specific instances of this general problem, for example, the case when H has degree at most one (monomer-dimer problem). In this paper, we present the first general subcase of the subgraph isomorphism counting problem which is almost always efficiently approximable. The results rely on a new graph decomposition technique. Informally, the decomposition is a labeling of the vertices such that every edge is between vertices with different labels and for every vertex all neighbors with a higher label have identical labels. The labeling implicitly generates a sequence of bipartite graphs which permits us to break the problem of counting embeddings of large subgraphs into that of counting embeddings of small subgraphs. Using this method, we present a simple randomized algorithm for the counting problem. For all decomposable graphs H and all graphs G, the algorithm is an unbiased estimator. Furthermore, for all graphs H having a decomposition where each of the bipartite graphs generated is small and almost all graphs G, the algorithm is a fully polynomial randomized approximation scheme. We show that the graph classes of H for which we obtain a fully polynomial randomized approximation scheme for almost all G includes graphs of degree at most two, bounded-degree forests, bounded-length grid graphs, subdivision of bounded-degree graphs, and major subclasses of outerplanar graphs, series-parallel graphs and planar graphs, whereas unbounded-length grid graphs are excluded.Comment: Earlier version appeared in Random 2008. Fixed an typo in Definition 3.

Enumerating Subgraph Instances Using Map-Reduce

The theme of this paper is how to find all instances of a given "sample" graph in a larger "data graph," using a single round of map-reduce. For the simplest sample graph, the triangle, we improve upon the best known such algorithm. We then examine the general case, considering both the communication cost between mappers and reducers and the total computation cost at the reducers. To minimize communication cost, we exploit the techniques of (Afrati and Ullman, TKDE 2011)for computing multiway joins (evaluating conjunctive queries) in a single map-reduce round. Several methods are shown for translating sample graphs into a union of conjunctive queries with as few queries as possible. We also address the matter of optimizing computation cost. Many serial algorithms are shown to be "convertible," in the sense that it is possible to partition the data graph, explore each partition in a separate reducer, and have the total computation cost at the reducers be of the same order as the computation cost of the serial algorithm.Comment: 37 page

Hamilton cycles in dense vertex-transitive graphs

A famous conjecture of Lov\'asz states that every connected vertex-transitive graph contains a Hamilton path. In this article we confirm the conjecture in the case that the graph is dense and sufficiently large. In fact, we show that such graphs contain a Hamilton cycle and moreover we provide a polynomial time algorithm for finding such a cycle.Comment: 26 pages, 3 figures; referees' comments incorporated; accepted for publication in Journal of Combinatorial Theory, series

An extensive English language bibliography on graph theory and its applications, supplement 1

Graph theory and its applications - bibliography, supplement