arXiv subjects

Bhaswar B. Bhattacharya

Publications and source records attributed to Bhaswar B. Bhattacharya.

60 records · Page 4Linked to original sources

arXiv2014

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

This paper proves limit theorems for the number of monochromatic edges in uniform random colorings of general random graphs. These can be seen as generalizations of the birthday problem (what is the chance that there are two friends with the same birthday?). It is shown that if the number of colors grows to infinity, the asymptotic distribution is either a Poisson mixture or a Normal depending solely on the limiting behavior of the ratio of the number of edges in the graph and the number of colors. This result holds for any graph sequence, deterministic or random. On the other hand, when the number of colors is fixed, a necessary and sufficient condition for asymptotic normality is determined. Finally, using some results from the emerging theory of dense graph limits, the asymptotic (non-normal) distribution is characterized for any converging sequence of dense graphs. The proofs are based on moment calculations which relate to the results of Erd\H os and Alon on extremal subgraph counts. As a consequence, a simpler proof of a result of Alon, estimating the number of isomorphic copies of a cycle of given length in graphs with a fixed number of edges, is presented.

math.PR↗

arXiv2013

On Pseudo-Convex Partitions of a Planar Point Set

Aichholzer et al. [{\it Graphs and Combinatorics}, Vol. 23, 481-507, 2007] introduced the notion of pseudo-convex partitioning of planar point sets and proved that the pseudo-convex partition number $ψ(n)$ satisfies, $\frac{3}{4}\lfloor\frac{n}{4}\rfloor\leq ψ(n)\leq\lceil\frac{n}{4}\rceil$. In this paper we prove that $ψ(13)=3$, which immediately improves the upper bound on $ψ(n)$ to $\lceil\frac{3n}{13}\rceil$, thus answering a question posed by Aichholzer et al. in the same paper.

math.CO↗

arXiv2012

Minimum-Weight Edge Discriminator in Hypergraphs

In this paper we introduce the concept of minimum-weight edge-discriminators in hypergraphs, and study its various properties. For a hypergraph $\mathcal H=(\mathcal V, \mathcal E)$, a function $λ: \mathcal V\rightarrow \mathbb Z^{+}\cup\{0\}$ is said to be an {\it edge-discriminator} on $\mathcal H$ if $\sum_{v\in E_i}{λ(v)}>0$, for all hyperedges $E_i\in \mathcal E$, and $\sum_{v\in E_i}{λ(v)}\ne \sum_{v\in E_j}{λ(v)}$, for every two distinct hyperedges $E_i, E_j \in \mathcal E$. An {\it optimal edge-discriminator} on $\mathcal H$, to be denoted by $λ_\mathcal H$, is an edge-discriminator on $\mathcal H$ satisfying $\sum_{v\in \mathcal V}λ_\mathcal H (v)=\min_λ\sum_{v\in \mathcal V}{λ(v)}$, where the minimum is taken over all edge-discriminators on $\mathcal H$. We prove that any hypergraph $\mathcal H=(\mathcal V, \mathcal E)$, with $|\mathcal E|=n$, satisfies $\sum_{v\in \mathcal V} λ_\mathcal H(v)\leq n(n+1)/2$, and equality holds if and only if the elements of $\mathcal E$ are mutually disjoint. For $r$-uniform hypergraphs $\mathcal H=(\mathcal V, \mathcal E)$, it follows from results on Sidon sequences that $\sum_{v\in \mathcal V}λ_{\mathcal H}(v)\leq |\mathcal V|^{r+1}+o(|\mathcal V|^{r+1})$, and the bound is attained up to a constant factor by the complete $r$-uniform hypergraph. Next, we construct optimal edge-discriminators for some special hypergraphs, which include paths, cycles, and complete $r$-partite hypergraphs. Finally, we show that no optimal edge-discriminator on any hypergraph $\mathcal H=(\mathcal V, \mathcal E)$, with $|\mathcal E|=n (\geq 3)$, satisfies $\sum_{v\in \mathcal V} λ_\mathcal H (v)=n(n+1)/2-1$, which, in turn, raises many other interesting combinatorial questions.

math.CO↗

arXiv2012

On the Fermat-Weber Point of a Polygonal Chain

In this paper, we study the properties of the Fermat-Weber point for a set of fixed points, whose arrangement coincides with the vertices of a regular polygonal chain. A $k$-chain of a regular $n$-gon is the segment of the boundary of the regular $n$-gon formed by a set of $k(\leq n)$ consecutive vertices of the regular $n$-gon. We show that for every odd positive integer $k$, there exists an integer $N(k)$, such that the Fermat-Weber point of a set of $k$ fixed points lying on the vertices a $k$-chain of a $n$-gon coincides with a vertex of the chain whenever $n\geq N(k)$. We also show that $\lceilπm(m+1)-π^2/4\rceil \leq N(k) \leq \lfloorπm(m+1)+1\rfloor$, where $k (=2m+1)$ is any odd positive integer. We then extend this result to a more general family of point set, and give an $O(hk\log k)$ time algorithm for determining whether a given set of $k$ points, having $h$ points on the convex hull, belongs to such a family.

math.MG↗

arXiv2012

Holes or Empty Pseudo-Triangles in Planar Point Sets

Let $E(k, \ell)$ denote the smallest integer such that any set of at least $E(k, \ell)$ points in the plane, no three on a line, contains either an empty convex polygon with $k$ vertices or an empty pseudo-triangle with $\ell$ vertices. The existence of $E(k, \ell)$ for positive integers $k, \ell\geq 3$, is the consequence of a result proved by Valtr [Discrete and Computational Geometry, Vol. 37, 565--576, 2007]. In this paper, following a series of new results about the existence of empty pseudo-triangles in point sets with triangular convex hulls, we determine the exact values of $E(k, 5)$ and $E(5, \ell)$, and prove bounds on $E(k, 6)$ and $E(6, \ell)$, for $k, \ell\geq 3$. By dropping the emptiness condition, we define another related quantity $F(k, \ell)$, which is the smallest integer such that any set of at least $F(k, \ell)$ points in the plane, no three on a line, contains a convex polygon with $k$ vertices or a pseudo-triangle with $\ell$ vertices. Extending a result of Bisztriczky and Tóth [Discrete Geometry, Marcel Dekker, 49--58, 2003], we obtain the exact values of $F(k, 5)$ and $F(k, 6)$, and obtain non-trivial bounds on $F(k, 7)$.

math.CO↗

arXiv2011

Disjoint Empty Convex Pentagons in Planar Point Sets

Harborth [{\it Elemente der Mathematik}, Vol. 33 (5), 116--118, 1978] proved that every set of 10 points in the plane, no three on a line, contains an empty convex pentagon. From this it follows that the number of disjoint empty convex pentagons in any set of $n$ points in the plane is least $\lfloor\frac{n}{10}\rfloor$. In this paper we prove that every set of 19 points in the plane, no three on a line, contains two disjoint empty convex pentagons. We also show that any set of $2m+9$ points in the plane, where $m$ is a positive integer, can be subdivided into three disjoint convex regions, two of which contains $m$ points each, and another contains a set of 9 points containing an empty convex pentagon. Combining these two results, we obtain non-trivial lower bounds 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 $\lfloor\frac{5n}{47}\rfloor$. This bound has been further improved to $\frac{3n-1}{28}$ for infinitely many $n$.

math.CO↗