Soumen Nandi

• PhD
• Indian Statistical Institute

An (n, m)-graph is a graph with n types of arcs and m types of edges. A homomorphism of an (n, m)-graph G to another (n, m)-graph H is a vertex mapping that preserves the adjacencies along with their types and directions. The order of a smallest (with respect to the number of vertices) such H is the (n, m)-chromatic number of G. Moreover, an (n, m)...

{\sc Sprout} is a two-player pen and paper game which starts with $n$ vertices, and the players take turns to join two pre-existing dots by a subdivided edge while keeping the graph sub-cubic planar at all times. The first player not being able to move loses. A major conjecture claims that Player 1 has a winning strategy if and only if $n \eq...

An $(n,m)$-graph is a graph with $n$ types of arcs and $m$ types of edges. A homomorphism of an $(n,m)$-graph $G$ to another $(n,m)$-graph $H$ is a vertex mapping that preserves adjacency, its direction, and its type. The minimum value of $|V(H)|$ such that $G$ admits a homomorphism to $H$ is the $(n,m)$-chromatic number of $G$, denoted by $\mychi_...

An $(n,m)$-graph is a graph with $n$ types of arcs and $m$ types of edges. A homomorphism of an $(n,m)$-graph $G$ to another $(n,m)$-graph $H$ is a vertex mapping that preserves the adjacencies along with their types and directions. The order of a smallest (with respect to the number of vertices) such $H$ is the $(n,m)$-chromatic number of $G$.More...

The radio k-chromatic number \(rc_k(G)\) of a graph G is the minimum integer \(\lambda \) such that there exists a function \(\phi : V(G) \rightarrow \{0,1,\cdots , \lambda \}\) satisfying \(|\phi (u)-\phi (v)| \ge k+1 - d(u,v)\), where d(u, v) denotes the distance between u and v. To date, several upper and lower bounds of \(rc_k(\cdot )\) is esta...

A proper $k$-coloring of a graph $G$ is a \emph{neighbor-locating $k$-coloring} if for each pair of vertices in the same color class, the sets of colors found in their neighborhoods are different. The neighbor-locating chromatic number $\chi_{NL}(G)$ is the minimum $k$ for which $G$ admits a neighbor-locating $k$-coloring. A proper $k$-coloring of...

A proper k-vertex-coloring of a graph G is a neighbor-locatingk-coloring if for each pair of vertices in the same color class, the sets of colors found in their neighborhoods are different. The neighbor-locating chromatic number χNL(G) is the minimum k for which G admits a neighbor-locating k-coloring. A proper k-vertex-coloring of a graph G is a l...

An (m,n)-colored mixed graph, or simply, an (m,n)-graph is a graph having m different types of arcs and n different types of edges. A homomorphism of an (m,n)-graph G to another (m,n)-graph H is a vertex mapping that preserves adjacency; and the type and direction of the adjacency. An (m,n)-relative clique of G is a vertex subset R whose images are...

In relation to oriented coloring and chromatic number, the parameter oriented relative clique number of an oriented graph $\overrightarrow{G}$, denoted by $\omega_{ro}(\overrightarrow{G})$, is the main focus of this work. We solve an open problem mentioned in the recent survey on oriented coloring by Sopena (Discrete Mathematics 2016), and positive...

A signed graph (G,σ) is a graph G along with a function σ:E(G)→{+,−}. A closed walk of a signed graph is positive (resp., negative) if it has an even (resp., odd) number of negative edges, counting repetitions. A homomorphism of a (simple) signed graph to another signed graph is a vertex-mapping that preserves adjacencies and signs of closed walks....

The radio $k$-coloring is an assignment $l$ of non-negative integers to the vertices of a graph in such a way that for two vertices $u,v$ which are $d$ distance apart, we must have $|l(u)-l(v)| \geq k+1 - d$. The difference between the greatest and the least image of $l$ is its span and our objective is to minimize the span. That is, the radio $k$-...

An (m,n)-colored mixed graph, or simply, an (m,n)-graph is a graph having m different types of arcs and n different types of edges. A homomorphism of an (m,n)-graph G to another (m,n)-graph H is a vertex mapping that preserves adjacency, the type thereto and the direction. A subset R of the set of vertices of G that always maps distinct vertices in...

Pushable homomorphisms and the pushable chromatic number χp of oriented graphs were introduced by Klostermeyer and MacGillivray in 2004. They notably observed that, for any oriented graph G⃗, we have χp(G⃗)≤χo(G⃗)≤2χp(G⃗), where χo(G⃗) denotes the oriented chromatic number of G⃗. This stands as the first general bounds on χp. This parameter was fur...

A signed graph $(G, \sigma)$ is a graph $G$ along with a function $\sigma: E(G) \to \{+,-\}$. A closed walk of a signed graph is positive (resp., negative) if it has an even (resp., odd) number of negative edges, counting repetitions. A homomorphism of a (simple) signed graph to another signed graph is a vertex-mapping that preserves adjacencies an...

Given a convex polygon with n vertices, we study the problem of identifying a triangle with its smallest side as large as possible among all the triangles that can be drawn inside the polygon. We show that at least one of the vertices of such a triangle must coincide with a vertex of the polygon. We also propose an O(n2) time algorithm to compute s...

A vertex subset R of an oriented graph \(\overrightarrow{G}\) is a relative oriented clique if each pair of non-adjacent vertices of R is connected by a directed 2-path. The relative oriented clique number \(\omega _{ro}(\overrightarrow{G})\) of \(\overrightarrow{G}\) is the maximum value of |R| where R is a relative oriented clique of \(\overright...

Pushable homomorphisms and the pushable chromatic number $\chi_p$ of oriented graphs were introduced by Klostermeyer and MacGillivray in 2004. They notably observed that, for any oriented graph $\overrightarrow{G}$, we have $\chi_p(\overrightarrow{G}) \leq \chi_o(\overrightarrow{G}) \leq 2 \chi_p(\overrightarrow{G})$, where $\chi_o(\overrightarrow{...

In this paper, we consider the dynamic version of covering the convex hull of a point set P in ℝ ² by two congruent disks of minimum size. Here, the points can be added or deleted in the set P, and the objective is to maintain a data structure that, at any instant of time, can efficiently report two disks of minimum size whose union completely cove...

An error is spotted in the statement of Theorem 1.3 of our published article titled “On oriented cliques with respect to push operation” (Discrete Applied Mathematics 2017). The theorem provided an exhaustive list of 16 minimal (up to spanning subgraph inclusion) underlying planar push cliques. The error was that, one of the 16 graphs from the abov...

An $(m, n)$-colored mixed graph is a graph having arcs of $m$ different colors and edges of $n$ different colors. A graph homomorphism of an $(m, n$)-colored mixed graph $G$ to an $(m, n)$-colored mixed graph $H$ is a vertex mapping such that if $uv$ is an arc (edge) of color $c$ in $G$, then $f(u)f(v)$ is also an arc (edge) of color $c$. The ($m,...

An error is spotted in the statement of Theorem~1.3 of our published article titled "On oriented cliques with respect to push operation" (Discrete Applied Mathematics 2017). The theorem provided an exhaustive list of 16 minimal (up to spanning subgraph inclusion) underlying planar push cliques. The error was that, one of the 16 graphs from the abov...

Given a convex polygon with n vertices, we study the problem of identifying a triangle with its smallest side as large as possible among all the triangles that can be drawn inside the polygon. We show that at least one of the vertices of such a triangle must be common with a vertex of the polygon. Next we propose an \(O(n^2\log n)\) time algorithm...

An n-radiok-coloring of a graph G is a function l:V(G)→{0,1,…,n} satisfying the condition |l(u)−l(v)|≥k+1−d(u,v) for all distinct u,v∈V(G). The radiok-chromatic numberrck(G) of G is the minimum n such that G admits an n-radio k-coloring. We establish a general technique for computing the lower bound for rck(G) of a general graph G and derive a form...

An (m, n)-colored mixed graph G is a graph with its arcs having one of the m different colors and edges having one of the n different colors. A homomorphism f of an (m, n)-colored mixed graph G to an (m, n)-colored mixed graph H is a vertex mapping such that if uv is an arc (edge) of color c in G, then f(u)f(v) is an arc (edge) of color c in H. The...

An - labeling of a simple graph is a mapping such that when the distance between and is for . The labeling span of a graph is the minimum such that admits an - labeling. In this article, we prove a conjecture by Calamoneri (2013) by showing where is the infinite triangular lattice. We also show that but for all .

A signed graph $ (G, \Sigma)$ is a graph positive and negative ($\Sigma $ denotes the set of negative edges). To re-sign a vertex $v$ of a signed graph $ (G, \Sigma)$ is to switch the signs of the edges incident to $v$. If one can obtain $ (G, \Sigma')$ by re-signing some vertices of $ (G, \Sigma)$, then $ (G, \Sigma) \equiv (G, \Sigma')$. A signed...

Given a convex polygon $P$ with $n$ vertices, the two-center problem is to find two congruent closed disks of minimum radius such that they completely cover $P$. We propose an algorithm for this problem in the streaming setup, where the input stream is the vertices of the polygon in clockwise order. It produces a radius $r$ satisfying $r\leq2r_{opt...

To push a vertex $v$ of a directed graph $\overrightarrow{G}$ is to change the orientations of all the arcs incident with $v$. An oriented graph is a directed graph without any cycle of length at most 2. An oriented clique is an oriented graph whose non-adjacent vertices are connected by a directed 2-path. A push clique is an oriented clique that r...

To push a vertex $v$ of a directed graph $\overrightarrow{G}$ is to change the orientations of all the arcs incident with $v$. An oriented graph is a directed graph without any cycle of length at most 2. An oriented clique is an oriented graph whose non-adjacent vertices are connected by a directed 2-path. A push clique is an oriented clique that r...

An $(m,n)$-colored mixed graph $G$ is a graph with its arcs having one of the $m$ different colors and edges having one of the $n$ different colors. A homomorphism $f$ of an $(m,n)$-colored mixed graph $G$ to an $(m,n)$-colored mixed graph $H$ is a vertex mapping such that if $uv$ is an arc (edge) of color $c$ in $G$, then $f(u)f(v)$ is an arc (edg...

An labeling of a simple graph G is a mapping such that , for all , where is the length of the shortest path connecting u and v. The labeling span of a family of graphs is the minimum n for which each admits an labeling. For the family of all subgraphs of an infinite triangular lattice we provide upper and lower bounds of for general k and show that...