30708438
On the independence and chromatic numbers of random regular graphs
Abstract Let G r denote a random r -regular graph with vertex set {1, 2, …, n } and α ( G r ) and χ ( G r ) denote respectively its independence and chromatic numbers. We show that with probability going to 1 as n → ∞ respectively |δ(G r ) − 2n r ( log r − log log r + 1 − log 2)|⩽ γn r and |χ(G r ) − r 2 log r − 8r log log r ( log ) 2 | ⩽ 8r log log r ( log r ) 2 provided r = o ( n θ ), θ 1 3 , 0 r ≥ r e , where r e depends on e only.
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.
27799444
Edge-colouring random graphs
Let pn denote the proportion of n-vertex graphs in class 2, that is, such that the chromatic index exceeds the maximum vertex degree. We extend a result of Erdős and Wilson, and show that n−(12 + e)n < pn < n−(18 − e)n for n sufficiently large.
19308721
Maximum matchings in a class of random graphs
Let Vn = {1,2,…,n} and D(n, m) be the set of digraphs with vertex set Vn in which each v ϵ Vn has outdegree m. D(n, m) is chosen uniformly at random from D(n, m) and then D(n, m) is obtained by ignoring the orientation of the edges of D(n, m). We show that 
Limn → ∞n evenPr(D(n, 1) has a perfect matching) = 0 
, 
Limn → ∞n evenPr(D(n, 2) has a perfect matching) = 1 
.
36434433
Hamiltonian cycles in random regular graphs
The study of the properties of random graphs was initiated by Erdos and RCnyi [2]. There they considered the asymptotic properties of random graphs with n labelled vertices and M(n) edges as n -+ 00. One problem they left open was the number of edges needed to ensure the almost certain existence of a Hamiltonian cycle. This was later partially solved by Posa [8] and then completely solved by Korshunov [6] and Komlos and Szemeridi [5]. The corresponding result for digraphs follows from a theorem of McDiarmid [7]. An interesting feature of this result, highlighted in [5], is that Hamiltonian cycles “appear” when M(n) is large enough to “ensure” that the graph has no vertices of degree one or zero. It is therefore of interest to study classes of graphs in which the minimal vertex degree exceeds some prescribed value. In [4] we studied random graphs obtained by taking a random digraph in which each vertex has outdegree m and then ignoring the orientation of the arcs. We showed that there exists mO m, is kept fixed and n + a, almost all such graphs have Hamiltonian cycles. We conjecture that the minimum value for m, is 3. In this paper we consider random vertex-labelled regular graphs of degree r and prove a similar result, viz that there exists r,, Q 796 such that, if r > rO 103
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.
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.