13374053
The Cover Time of Random Regular Graphs
Let $r \ge 3$ be constant, and let ${\cal G}_{r}$ denote the set of r-regular graphs with vertex set V = {1,2,...,n}. Let G be chosen randomly from ${\cal G}_{r}$. We prove that with high probability (\whp) the cover time of a random walk on G is asymptotic to $\frac{r-1}{r-2}\;n\log n$.
2168830
Generating and Counting Hamilton Cycles in Random Regular Graphs
LetGbe chosen uniformly at random from the set G(r,n) ofr-regular graphs with vertex set n. We describe polynomial time algorithms thatwhp(i) find a Hamilton cycle inG, and (ii) approximately count the number of Hamilton cycles inG.
12475992
On Markov Chains for Randomly H-Coloring a Graph
Let H=(W,F) be a graph without multiple edges, but with the possibility of having loops. Let G=(V,E) be a simple graph. A homomorphism c is a map c:V?W with the property that (v,w)?E implies that (c(v),c(w))?F. We will often refer to c(v) as the color of v and c as an H-coloring of G. We consider the problem of choosing a random H-coloring of G by Markov chain Monte Carlo. The probabilistic model we consider includes random proper colorings, random independent sets, and the Widom?Rowlinson and Beach models of statistical physics. We prove negative results for uniform sampling and a positive result for weighted sampling when H is a tree.
6616465
Multi-Coloured Hamilton Cycles In Random Edge-Coloured Graphs
We define a space of random edge-coloured graphs Gn,m,κ which correspond naturally to edge κ-colourings of Gn,m. We show that there exist constants K0, K1 ≤ 21 such that, provided m ≥ K0n log n and κ ≥ K1n, then a random edge-coloured graph contains a multi-coloured Hamilton cycle with probability tending to 1 as the number of vertices n tends to infinity.
6378032
Adding random edges to dense graphs
This paper investigates the addition of random edges to arbitrary dense graphs; in particular, we determine the number of random edges required to ensure various monotone properties including the appearance of a fixed size clique, small diameter and k-connectivity.
1964724
Optimal construction of edge-disjoint paths in random regular graphs
Given a graph G = (V, E) and a set of κ pairs of vertices in V, we are interested in finding, for each pair (ai, bi), a path connecting ai to bi such that the set of κ paths so found is edge-disjoint. (For arbitrary graphs the problem is NP-complete, although it is in P if κ is fixed.)We present a polynomial time randomized algorithm for finding edge-disjoint paths in the random regular graph Gn,r, for sufficiently large r. (The graph is chosen first, then an adversary chooses the pairs of end-points.) We show that almost every Gn,r is such that all sets of κ = Ω(n/log n) pairs of vertices can be joined. This is within a constant factor of the optimum.
5329008
Clustering in large graphs and matrices
We consider the problem of dividing a set of m points in Euclidean n-space into k clusters (m, n are variable while k is fixed), so as to minimize the sum of distances squared of each point to its “cluster center”. This formulation differs in two ways from the most frequently considered clustering problems in the literature, namely, here we have k fixed and m, n variable, and we use the sum of squared distances as our measure; we will argue that our problem is natural in many contexts. We consider a relaxation of the discrete problem: find the k-dimensional subspace V so that the sum of distances squared to V (of the m points) is minimized. We show: (i) The relaxation can be solved by the Singular Value Decomposition (SVD) of Linear Algebra. (ii) The solution of the relaxation can be used to get a 2-approximation algorithm for the original problem. More importantly, (iii) we argue that in fact the relaxation provides a generalized clustering which is useful in its own right. Finally, (iv) we show that the SVD of a randomly chosen submatrix (according to a suitable probability distribution) of the matrix provides an approximation to the SVD of the whole matrix, thus yielding a very fast randomized algorithm. This cam be applied to problems of very large size which typically arise in modern applications.