Bhaswar B. Bhattacharya

Upper tails and independence polynomials in random graphs

Abstract The upper tail problem in the Erdős–Renyi random graph G ∼ G n , p asks to estimate the probability that the number of copies of a graph H in G exceeds its expectation by a factor 1 + δ . Chatterjee and Dembo showed that in the sparse regime of p → 0 as n → ∞ with p ≥ n − α for an explicit α = α H > 0 , this problem reduces to a natural variational problem on weighted graphs, which was thereafter asymptotically solved by two of the authors in the case where H is a clique. Here we extend the latter work to any fixed graph H and determine a function c H ( δ ) such that, for p as above and any fixed δ > 0 , the upper tail probability is exp ⁡ [ − ( c H ( δ ) + o ( 1 ) ) n 2 p Δ log ⁡ ( 1 / p ) ] , where Δ is the maximum degree of H. As it turns out, the leading order constant in the large deviation rate function, c H ( δ ) , is governed by the independence polynomial of H, defined as P H ( x ) = ∑ i H ( k ) x k where i H ( k ) is the number of independent sets of size k in H. For instance, if H is a regular graph on m vertices, then c H ( δ ) is the minimum between 1 2 δ 2 / m and the unique positive solution of P H ( x ) = 1 + δ .

Exact and Asymptotic Results on Coarse Ricci Curvature of Graphs

In this paper we study Olliviers coarse Ricci-curvature for graphs, and obtain exact formulas for the Ricci-curvature for bipartite graphs and for the graphs with girth at least 5. These are the first formulas for Ricci-curvature which hold for a wide class of graphs. We also obtain a general lower bound on the Ricci-curvature involving the size of the maximum matching in an appropriate subgraph. As a consequence, we characterize Ricci-flat graphs of girth 5, and give the first necessary and sufficient condition for the structure of Ricci-flat regular graphs of girth 4. Finally, we obtain the asymptotic Ricci-curvature of random bipartite graphs

New variations of the maximum coverage facility location problem

G(n,n, p)

Disjoint empty convex pentagons in planar point sets

and random graphs

Maximizing Voronoi Regions of a Set of Points Enclosed in a Circle with Applications to Facility Location

G(n, p)

Inference in Ising models

, in various regimes of

Universal Limit Theorems in Graph Coloring Problems With Connections to Extremal Combinatorics

p

DEGREE SEQUENCE OF RANDOM PERMUTATION GRAPHS

.

Almost empty monochromatic triangles in planar point sets

Consider a competitive facility location scenario where, given a set U of n users and a set F of m facilities in the plane, the objective is to place a new facility in an appropriate place such that the number of users served by the new facility is maximized. Here users and facilities are considered as points in the plane, and each user takes service from its nearest facility, where the distance between a pair of points is measured in either L1 or L2 or L∞ metric. This problem is also known as the maximum coverage (MaxCov) problem. In this paper, we will consider the k-MaxCov problem, where the objective is to place k (⩾1) new facilities such that the total number of users served by these k new facilities is maximized. We begin by proposing an O(nlogn) time algorithm for the k-MaxCov problem, when the existing facilities are all located on a single straight line and the new facilities are also restricted to lie on the same line. We then study the 2-MaxCov problem in the plane, and propose an O(n2) time and space algorithm in the L1 and L∞ metrics. In the L2 metric, we solve the 2-MaxCov problem in the plane in O(n3logn) time and O(n2logn) space. Finally, we consider the 2-Farthest-MaxCov problem, where a user is served by its farthest facility, and propose an algorithm that runs in O(nlogn) time, in all the three metrics.

Collision times in multicolor urn models and sequential graph coloring with applications to discrete logarithms

AbstractIn this paper we obtain the first non-trivial lower bound on the number of disjoint empty convex pentagons in planar points sets. We show that the number of disjoint empty convex pentagons in any set of n points in the plane, no three on a line, is at least