Abstract not found

We consider a canonical Ramsey type problem. An edge-coloring of a graph is called m-good if each color appears at most m times at each vertex. Fixing a graph G and a positive integer m, let f(m, G) denote the smallest n such that every m-good edge-coloring of K n yields a properly edge-colored copy of G, and let g(m, G) denote the smallest n such that every m-good edgecoloring of K n yields a rainbow copy of G. We give bounds on f(m, G) and g(m, G). For complete graphs G � K t, we have c 1mt 2 /ln t � f(m, K t) � c 2mt 2, and c � 1mt 3 /ln t � g(m, K t) � c � 2mt 3 /ln t, where c 1, c 2, c � 1, c � 2 are absolute constants. We also give bounds on f(m, G) and g(m, G) for general graphs G in terms of degrees in G. In particular, we show that for fixed m and d, and all sufficiently large n compared to m and d, f(m, G) � n for all graphs G with n vertices and

In this paper we study a random graph with N nodes, where node j has degree Dj and {Dj} N j=1 are i.i.d. with P(Dj ≤ x) = F(x). We assume that 1 − F(x) ≤ cx −τ+1 for some τ> 3 and some constant c> 0. This graph model is a variant of the so-called configuration model, and includes heavy tail degrees with finite variance. The minimal number of edges between two arbitrary connected nodes, also known as the graph distance or the hopcount, is investigated when N → ∞. We prove that the graph distance grows like log ν N, when the base of the logarithm equals ν = E[Dj(Dj − 1)]/E[Dj]> 1. This confirms the heuristic argument of Newman, Strogatz and Watts [35]. In addition, the random fluctuations around this asymptotic mean log ν N are characterized and shown to be uniformly bounded. In particular, we show convergence in distribution of the centered graph distance along exponentially growing subsequences. 1

Abstract not found

We prove that any collection of n discs in which each one intersects at most k others, can be colored with at most O(log 3 k) colors so that for each point p in the union of all discs there is at least one disc in the collection containing p whose color differs from that of all other members of the collection that contain p. This is motivated by a problem on frequency assignments in cellular networks, and improves the best previously known upper bound of O(log n) when k is much smaller than n. 1

The problem of L multiple descriptions of a stationary and ergodic Gaussian source with two levels of receivers is investigated. Each of the first level receivers receive (an arbitrary subset) k of the L descriptions, (k < L). The second level receiver receives all L descriptions. All the receivers, both at the first level and the second level, reconstruct the source using the subset of descriptions they receive. The corresponding reconstructions are subject to quadratic distortion constraints. Our main result is the derivation of an outer bound on the sum rate of the descriptions so that the distortion constraints are met. We show that a natural analog-digital separation architecture involving joint Gaussian vector quantizers and a binning scheme meets this outer bound with equality for several scenarios. These scenarios include the case when the distortion constraints are symmetric and the case for general distortion constraints with k = 2 and L = 3. We also show the robustness of this architecture: the distortions achieved are no larger when used to describe any non-Gaussian source with the same covariance matrix. 1

Abstract. Let C1 denote the largest connected component of the critical Erdős-Rényi random graph G(n, 1). We show that, typically, the diameter of C1 is of n order n 1/3 and the mixing time of the lazy simple random walk on C1 is of order n. The latter answers a question of Benjamini, Kozma and Wormald [5]. These results extend to clusters of size n 2/3 of p-bond percolation on any d-regular n-vertex graph where such clusters exist, provided that p(d − 1) ≤ 1 + O(n −1/3). 1.

We provide an overview of an emerging area of domination analysis (DA) of combinatorial optimization algorithms and problems. We consider DA theory and its relevance to computational practice. 1

In this paper we prove the following result about vertex list colourings, which shows that a conjecture from [9] is asymptotically correct. Let G be a graph with the sets of lists S(v), satisfying that for every vertex jS(v)j = (1+o(1))d and for each colour c 2 S(v), the number of neighbours of v that have c in their list is at most d. Then there exist a proper colouring of G from these lists.

We describe the component sizes in critical independent p-bond percolation on a random d-regular graph on n vertices, where d ≥ 3 is fixed and n grows. We prove mean-field behavior around the critical probability pc = 1 d−1. In particular, we show that there is a scaling window of width n −1/3 around pc in which the sizes of the largest components are roughly n 2/3 and we describe their limiting joint distribution. We also show that for the subcritical regime, i.e. p = (1 − ε(n))pc where ε(n) = o(1) but ε(n)n 1/3 → ∞, the sizes of the largest components are concentrated around an explicit function of n and ε(n) which is of order o(n 2/3). In the supercritical regime, i.e. p = (1 + ε(n))pc where ε(n) = o(1) but ε(n)n1/3 → ∞, the size of the largest component is concentrated around the value 2d d−2ε(n)n and a duality principle holds: other component sizes are distributed as in the subcritical regime.