Deepak Rajendraprasad

Bio: Deepak Rajendraprasad is an academic researcher from Indian Institutes of Technology. The author has contributed to research in topics: Chordal graph & Dimension (graph theory). The author has an hindex of 13, co-authored 67 publications receiving 513 citations. Previous affiliations of Deepak Rajendraprasad include University of Haifa & Indian Institute of Science.

Oriented Diameter of Star Graphs

Abstract: An orientation of an undirected graph G is an assignment of exactly one direction to each edge of G. Converting two-way traffic networks to one-way traffic networks and bidirectional communication networks to unidirectional communication networks are practical instances of graph orientations. In these contexts minimising the diameter of the resulting oriented graph is of prime interest.

On Coupon Coloring of Cartesian Product of Some Graphs

Abstract: Let G be a graph with no isolated vertices A k-coupon coloring of G is an assignment of k colors to the vertices of G such that every vertex contains vertices of all k colors in its neighborhood The coupon chromatic number of G, denoted \(\chi _c (G)\), is the maximum k for which a k-coupon coloring exists In this paper, we present an upper bound for the coupon chromatic number of Cartesian product of graphs G and H in terms of |V(G)| and |V(H)| Further, we prove that if G and H are bipartite graphs then \(G\square H\) has a coupon coloring with \(2\min \{\chi _c(G),\chi _c(H)\}\) colors As consequences, for any positive integer n, we obtain the coupon chromatic number and total domination number of n-dimensional torus \(\mathop {\square }\limits _{i=1}^{n} C_{k_i}\) with some suitable conditions to each \(k_i\), which turns out to be a generalization of the result due to S Gravier [Total domination number of grid graphs, Discrete Appl Math 121 (2002) 119–128] Finally, for any \(r\ge 0, d\ge 2\), we obtain the coupon coloring for the Hamming graph \(K_d\square K_d\square \cdots \square K_d\) (\(2^r\) times)

New bounds on the Ramsey number $r(I_m, L_n)$

Abstract: We investigate the Ramsey numbers $r(I_m, L_n)$ which is the minimal natural number $k$ such that every oriented graph on $k$ vertices contains either an independent set of size $m$ or a transitive tournament on $n$ vertices. Apart from the finitary combinatorial interest, these Ramsey numbers are of interest to set theorists since it is known that $r(\omega m, n) = \omega r(I_m, L_n)$, where $\omega$ is the lowest transfinite ordinal number, and $r(\kappa m, n) = \kappa r(I_m, L_n)$ for all initial ordinals $\kappa$. Continuing the research by Bermond from 1974 who did show $r(I_3, L_3) = 9$, we prove $r(I_4, L_3) = 15$ and $r(I_5, L_3) = 23$. The upper bounds for both the estimates above are obtained by improving the upper bound of $m^2$ on $r(I_m, L_3)$ due to Larson and Mitchell (1997) to $m^2 - m + 3$. Additionally, we provide asymptotic upper bounds on $r(I_m, L_n)$ for all $n \geq 3$. In particular, we show that $r(I_m, L_3) \in \Theta(m^2 / \log m)$.

Disjoint Total Dominating Sets in Near-Triangulations

Abstract: We show that every simple planar near-triangulation with minimum degree at least three contains two disjoint total dominating sets. The class includes all simple planar triangulations other than the triangle. This affirms a conjecture of Goddard and Henning [Thoroughly dispersed colorings, J. Graph Theory, 88 (2018) 174–191].

Partial list colouring of certain graphs

Abstract: Let $G$ be a graph on $n$ vertices and let $\mathcal{L}_k$ be an arbitrary function that assigns each vertex in $G$ a list of $k$ colours. Then $G$ is $\mathcal{L}_k$-list colourable if there exists a proper colouring of the vertices of $G$ such that every vertex is coloured with a colour from its own list. We say $G$ is $k$-choosable if for every such function $\mathcal{L}_k$, $G$ is $\mathcal{L}_k$-list colourable. The minimum $k$ such that $G$ is $k$-choosable is called the list chromatic number of $G$ and is denoted by $\chi_L(G)$. Let $\chi_L(G) = s$ and let $t$ be a positive integer less than $s$. The partial list colouring conjecture due to Albertson et al. \cite{albertson2000partial} states that for every $\mathcal{L}_t$ that maps the vertices of $G$ to $t$-sized lists, there always exists an induced subgraph of $G$ of size at least $\frac{tn}{s}$ that is $\mathcal{L}_t$-list colourable. In this paper we show that the partial list colouring conjecture holds true for certain classes of graphs like claw-free graphs, graphs with large chromatic number, chordless graphs, and series-parallel graphs. In the second part of the paper, we put forth a question which is a variant of the partial list colouring conjecture: does $G$ always contain an induced subgraph of size at least $\frac{tn}{s}$ that is $t$-choosable? We show that the answer to this question is not always `yes' by explicitly constructing an infinite family of $3$-choosable graphs where a largest induced $2$-choosable subgraph of each graph in the family is of size at most $\frac{5n}{8}$.

Property Testing and its connection to Learning and Approximation

Abstract: In this paper, we consider the question of determining whether a function f has property P or is e-far from any function with property P. A property testing algorithm is given a sample of the value of f on instances drawn according to some distribution. In some cases, it is also allowed to query f on instances of its choice. We study this question for different properties and establish some connections to problems in learning theory and approximation.In particular, we focus our attention on testing graph properties. Given access to a graph G in the form of being able to query whether an edge exists or not between a pair of vertices, we devise algorithms to test whether the underlying graph has properties such as being bipartite, k-Colorable, or having a p-Clique (clique of density p with respect to the vertex set). Our graph property testing algorithms are probabilistic and make assertions that are correct with high probability, while making a number of queries that is independent of the size of the graph. Moreover, the property testing algorithms can be used to efficiently (i.e., in time linear in the number of vertices) construct partitions of the graph that correspond to the property being tested, if it holds for the input graph.

Rainbow connections of graphs -- A survey

Abstract: The concept of rainbow connection was introduced by Chartrand et al. in 2008. It is fairly interesting and recently quite a lot papers have been published about it. In this survey we attempt to bring together most of the results and papers that dealt with it. We begin with an introduction, and then try to organize the work into five categories, including (strong) rainbow connection number, rainbow $k$-connectivity, $k$-rainbow index, rainbow vertex-connection number, algorithms and computational complexity. This survey also contains some conjectures, open problems or questions.

Rainbow Connections of Graphs: A Survey

Abstract: The concept of rainbow connection was introduced by Chartrand et al. [14] in 2008. It is interesting and recently quite a lot papers have been published about it. In this survey we attempt to bring together most of the results and papers that dealt with it. We begin with an introduction, and then try to organize the work into five categories, including (strong) rainbow connection number, rainbow k-connectivity, k-rainbow index, rainbow vertex-connection number, algorithms and computational complexity. This survey also contains some conjectures, open problems and questions.

Improved Testing Algorithms for Monotonicity.

Abstract: We present improved algorithms for testing monotonicity of functions. Namely, given the ability to query an unknown function f: Σ n ↦ Ξ, where Σ and Ξ are finite ordered sets, the test always accepts a monotone f, and rejects f with high probability if it is e-far from being monotone (i.e., every monotone function differs from f on more than an e fraction of the domain). For any e > 0, the query complexity of the test is O((n/e) · log ∣Σ ∣ · log ∣Ξ∣). The previous best known bound was \(\tilde{O}((n^2/\epsilon) \cdot \vert\Sigma\vert^2 \cdot \vert\Xi\vert)\).

Property Testing Lower Bounds via Communication Complexity.

Abstract: We develop a new technique for proving lower bounds in property testing, by showing a strong connection between testing and communication complexity. We give a simple scheme for reducing communication problems to testing problems, thus allowing us to use known lower bounds in communication complexity to prove lower bounds in testing. This scheme is general and implies a number of new testing bounds, as well as simpler proofs of several known bounds. For the problem of testing whether a boolean function is k-linear (a parity function on k variables), we achieve a lower bound of Omega(k) queries, even for adaptive algorithms with two-sided error, thus confirming a conjecture of Goldreich (2010). The same argument behind this lower bound also implies a new proof of known lower bounds for testing related classes such as k-juntas. For some classes, such as the class of monotone functions and the class of s-sparse GF(2) polynomials, we significantly strengthen the best known bounds.