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.

Partial List Colouring of Certain Graphs

Abstract: The partial list colouring conjecture due to Albertson, Grossman, and Haas (2000) states that for every $s$-choosable graph $G$ and every assignment of lists of size $t$, $1 \leq t \leq s$, to the vertices of $G$ there is an induced subgraph of $G$ on at least $\frac{t|V(G)|}{s}$ vertices which can be properly coloured from these lists. In this paper, we show that the partial list colouring conjecture holds true for certain classes of graphs like claw-free graphs, graphs with chromatic number at least $\frac{|V(G)|-1}{2}$, chordless graphs, and series-parallel graphs.

Upper bound on cubicity in terms of boxicity for graphs of low chromatic number

Abstract: The boxicity (respectively cubicity) of a graph $G$ is the minimum non-negative integer $k$, such that $G$ can be represented as an intersection graph of axis-parallel $k$-dimensional boxes (respectively $k$-dimensional unit cubes) and is denoted by $box(G)$ (respectively $cub(G)$). It was shown by Adiga and Chandran (Journal of Graph Theory, 65(4), 2010) that for any graph $G$, $cub(G) \le$ box$(G) \left \lceil \log_2 \alpha \right \rceil$, where $\alpha = \alpha(G)$ is the cardinality of the maximum independent set in $G$. In this note we show that $cub(G) \le 2 \left \lceil \log_2 \chi(G) \right \rceil box(G) + \chi(G) \left \lceil \log_2 \alpha(G) \right \rceil $. In general, this result can provide a much better upper bound than that of Adiga and Chandran for graph classes with bounded chromatic number. For example, for bipartite graphs we get, $cub(G) \le 2 (box(G) + \left \lceil \log_2 \alpha(G) \right \rceil )$. Moreover we show that for every positive integer $k$, there exist graphs with chromatic number $k$, such that for every $\epsilon > 0$, the value given by our upper bound is at most $(1+\epsilon)$ times their cubicity. Thus, our upper bound is almost tight.

Dimension of CPT posets

Abstract: A containment model $M_{\mathcal{P}}$ of a poset $\mathcal{P}=(X,\preceq)$ maps every $x \in X$ to a set $M_x$ such that, for every distinct $x,y \in X,\ x \preceq y$ if and only if $M_x \varsubsetneq M_y$. We shall be using the collection $(M_x)_{x \in X}$ to identify the containment model $M_{\mathcal{P}}$. A poset $\mathcal{P}=(X,\preceq)$ is a Containment order of Paths in a Tree (CPT poset), if it admits a containment model $M_{\mathcal{P}}=(P_x)_{x \in X}$ where every $P_x$ is a path of a tree $\mathcal{T}$, which is called the host tree of the model. In this paper, we give an asymptotically tight bound on the dimension of a CPT poset, which is tight up to a multiplicative factor of $(2+\epsilon)$, where $0 < \epsilon < 1$, with the help of a constructive proof. We show that if a poset $\mathcal{P}$ admits a CPT model in a host tree $\mathcal{T}$ of maximum degree $\Delta$ and radius $r$, then $dim(\mathcal{P}) \leq 2\lceil \log_{2}{\log_{2}{\Delta}} \rceil +2\lceil \log_{2}{r} \rceil+3$.

Approximation bounds on maximum edge 2-coloring of dense graphs

Abstract: For a graph $G$ and integer $q\geq 2$, an edge $q$-coloring of $G$ is an assignment of colors to edges of $G$, such that edges incident on a vertex span at most $q$ distinct colors. The maximum edge $q$-coloring problem seeks to maximize the number of colors in an edge $q$-coloring of a graph $G$. The problem has been studied in combinatorics in the context of {\em anti-Ramsey} numbers. Algorithmically, the problem is NP-Hard for $q\geq 2$ and assuming the unique games conjecture, it cannot be approximated in polynomial time to a factor less than $1+1/q$. The case $q=2$, is particularly relevant in practice, and has been well studied from the view point of approximation algorithms. A $2$-factor algorithm is known for general graphs, and recently a $5/3$-factor approximation bound was shown for graphs with perfect matching. The algorithm (which we refer to as the matching based algorithm) is as follows: "Find a maximum matching $M$ of $G$. Give distinct colors to the edges of $M$. Let $C_1,C_2,\ldots, C_t$ be the connected components that results when M is removed from G. To all edges of $C_i$ give the $(|M|+i)$th color." In this paper, we first show that the approximation guarantee of the matching based algorithm is $(1 + \frac {2} {\delta})$ for graphs with perfect matching and minimum degree $\delta$. For $\delta \ge 4$, this is better than the $\frac {5} {3}$ approximation guarantee proved in {AAAP}. For triangle free graphs with perfect matching, we prove that the approximation factor is $(1 + \frac {1}{\delta - 1})$, which is better than $5/3$ for $\delta \ge 3$.

Separation dimension of sparse graphs.

Abstract: The separation dimension of a graph $G$ is the smallest natural number $k$ for which the vertices of $G$ can be embedded in $\mathbb{R}^k$ such that any pair of disjoint edges in $G$ can be separated by a hyperplane normal to one of the axes. Equivalently, it is the smallest possible cardinality of a family $\mathcal{F}$ of permutations of the vertices of $G$ such that for any two disjoint edges of $G$, there exists at least one permutation in $\mathcal{F}$ in which all the vertices in one edge precede those in the other. In general, the maximum separation dimension of a graph on $n$ vertices is $\Theta(\log n)$. In this article, we focus on sparse graphs and show that the maximum separation dimension of a $k$-degenerate graph on $n$ vertices is $O(k \log\log n)$ and that there exists a family of $2$-degenerate graphs with separation dimension $\Omega(\log\log n)$. We also show that the separation dimension of the graph $G^{1/2}$ obtained by subdividing once every edge of another graph $G$ is at most $(1 + o(1)) \log\log \chi(G)$ where $\chi(G)$ is the chromatic number of the original graph.

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.