arXiv subjects

Soumen Nandi

Publications and source records attributed to Soumen Nandi.

15 recordsLinked to original sources

arXiv2024

On locating and neighbor-locating colorings of sparse graphs

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 two sets of colors found in their respective neighborhoods are different. The \textit{neighbor-locating chromatic number} $χ_{NL}(G)$ is the minimum $k$ for which $G$ admits a neighbor-locating $k$-coloring. A proper $k$-vertex-coloring of a graph $G$ is a \emph{locating $k$-coloring} if for each pair of vertices $x$ and $y$ in the same color-class, there exists a color class $S_i$ such that $d(x,S_i)\neq d(y,S_i)$. The locating chromatic number $χ_{L}(G)$ is the minimum $k$ for which $G$ admits a locating $k$-coloring. Our main results concern the largest possible order of a sparse graph of given neighbor-locating chromatic number. More precisely, we prove that if $G$ has order $n$, neighbor-locating chromatic number $k$ and average degree at most $2a$, where $2a\le k-1$ is a positive integer, then $n$ is upper-bounded by $\mathcal{O}(a^2(k^{2a+1}))$. We also design a family of graphs of bounded maximum degree whose order is close to reaching this upper bound. Our upper bound generalizes two previous bounds from the literature, which were obtained for graphs of bounded maximum degree and graphs of bounded cycle rank, respectively. Also, we prove that determining whether $χ_L(G)\le k$ and $χ_{NL}(G)\le k$ are NP-complete for sparse graphs: more precisely, for graphs with average degree at most 7, maximum average degree at most 20 and that are $4$-partite. We also study the possible relation between the ordinary chromatic number, the locating chromatic number and the neighbor-locating chromatic number of a graph.

math.CO↗

arXiv2024

A linear algorithm for radio $k$-coloring of powers of paths having small diameters

The radio $k$-chromatic number $rc_k(G)$ of a graph $G$ is the minimum integer $λ$ such that there exists a function $ϕ: V(G) \to \{0,1,\cdots, λ\}$ satisfying $|ϕ(u)-ϕ(v)| \geq k+1 - d(u,v)$, where $d(u,v)$ denotes the distance between $u$ and $v$. A considerable amount of attention has been given to find the exact values or providing polynomial time algorithms to determine $rc_k(G)$ for several basic graph families such as paths, cycles, trees, and powers of paths, usually for some specific values of $k$. In this article, we find the exact values of $rc_k(G)$ where $G$ is a power of a path with diameter strictly less than $k$. Our proof readily provides a linear time algorithm for assigning a radio $k$-coloring of $G$. Furthermore, our proof technique is a potential tool for solving the same problem for other classes of graphs having ``small'' diameters.

math.CO↗

arXiv2024

On $(n,m)$-chromatic numbers of graphs having bounded sparsity parameters

An $(n,m)$-graph is characterised by having $n$ types of arcs and $m$ types of edges. A homomorphism of an $(n,m)$-graph $G$ to an $(n,m)$-graph $H$, is a vertex mapping that preserves adjacency, direction, and type. The $(n,m)$-chromatic number of $G$, denoted by $χ_{n,m}(G)$, is the minimum value of $|V(H)|$ such that there exists a homomorphism of $G$ to $H$. The theory of homomorphisms of $(n,m)$-graphs have connections with graph theoretic concepts like harmonious coloring, nowhere-zero flows; with other mathematical topics like binary predicate logic, Coxeter groups; and has application to the Query Evaluation Problem (QEP) in graph database. In this article, we show that the arboricity of $G$ is bounded by a function of $χ_{n,m}(G)$ but not the other way around. Additionally, we show that the acyclic chromatic number of $G$ is bounded by a function of $χ_{n,m}(G)$, a result already known in the reverse direction. Furthermore, we prove that the $(n,m)$-chromatic number for the family of graphs with a maximum average degree less than $2+ \frac{2}{4(2n+m)-1}$, including the subfamily of planar graphs with girth at least $8(2n+m)$, equals $2(2n+m)+1$. This improves upon previous findings, which proved the $(n,m)$-chromatic number for planar graphs with girth at least $10(2n+m)-4$ is $2(2n+m)+1$. It is established that the $(n,m)$-chromatic number for the family $\mathcal{T}_2$ of partial $2$-trees is both bounded below and above by quadratic functions of $(2n+m)$, with the lower bound being tight when $(2n+m)=2$. We prove $14 \leq χ_{(0,3)}(\mathcal{T}_2) \leq 15$ and $14 \leq χ_{(1,1)}(\mathcal{T}_2) \leq 21$ which improves both known lower bounds and the former upper bound. Moreover, for the latter upper bound, to the best of our knowledge we provide the first theoretical proof.

math.CO↗

arXiv2023

A theoretical expansion of the Sprout game

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 $n \equiv 3,4,5$ ($\bmod~6$). The conjecture is verified until $44$, and a few isolated values of $n$, usually with the help of a computer. However, to the best of our understanding, not too much progress could be made towards finding a theoretical proof of the conjecture till now. In this article, we try to take a bottom-up approach and start building a theory around the problem. We start by expanding a related game called Brussels Sprout (where dots are replaced by crosses) introduced by Conway, possibly to help the understanding of Sprout. In particular, we introduce and study a generalized version of Brussels Sprout where crosses are replaced by a dot having an arbitrary number of ``partial edges'' (say, general cross) coming out, and planar graphs are replaced by any (pre-decided) hereditary class of graphs. We study the game for forests, graphs on surfaces, and sparse planar graphs. We also do a nimber characterization of the game when the hereditary class is taken to be triangle-free planar graphs, and we have started the game with two arbitrary generalized crosses. Moreover, while studying this particular case, we naturally stumble upon a circular version of the same game and solve a difficult nimber characterization using the method of structural induction. The above mentioned proof may potentially be one approach to solving the Sprout conjecture.

math.CO↗

arXiv2023

On coloring parameters of triangle-free planar $(n,m)$-graphs

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 $(n,m)$-relative clique $R$ of an $(n,m)$-graph $G$ is a vertex subset of $G$ for which no two distinct vertices of $R$ get identified under any homomorphism of $G$. The $(n,m)$-relative clique number of $G$, denoted by $ω_{r(n,m)}(G)$, is the maximum $|R|$ such that $R$ is an $(n,m)$-relative clique of $G$. In practice, $(n,m)$-relative cliques are often used for establishing lower bounds of $(n,m)$-chromatic number of graph families. Generalizing an open problem posed by Sopena [Discrete Mathematics 2016] in his latest survey on oriented coloring, Chakroborty, Das, Nandi, Roy and Sen [Discrete Applied Mathematics 2022] conjectured that $ω_{r(n,m)}(G) \leq 2 (2n+m)^2 + 2$ for any triangle-free planar $(n,m)$-graph $G$ and that this bound is tight for all $(n,m) \neq (0,1)$.In this article, we positively settle this conjecture by improving the previous upper bound of $ω_{r(n,m)}(G) \leq 14 (2n+m)^2 + 2$ to $ω_{r(n,m)}(G) \leq 2 (2n+m)^2 + 2$, and by finding examples of triangle-free planar graphs that achieve this bound. As a consequence of the tightness proof, we also establish a new lower bound of $2 (2n+m)^2 + 2$ for the $(n,m)$-chromatic number for the family of triangle-free planar graphs.

math.CO↗

arXiv2022

On homomorphism related parameters of oriented triangle-free planar graphs

The first major contribution of this work is proving that the oriented relative clique number of oriented triangle-free planar graphs is $10$, which completely answers and closes an open problem posed by Sopena (Discrete Mathematics 2016) in the most recent survey on oriented colorings. The second major contribution of the paper is to prove that if all oriented triangle-free planar graphs admit a homomorphism to a particular oriented graph $\overrightarrow{T}$, then its underlying graph $T$ must have minimum degree at least $10$. This result implies that, for the family of oriented triangle-free planar graphs, the lower bounds of the parameters oriented chromatic number, pushable chromatic number, $2$-dipath $L(p,1)$-labeling span, and oriented $L(p,1)$-labeling span are at least $11$, $6$, $p+8$, and $2p+8$, respectively, where $p \geq 1$. That is, we are able to obtain improved lower bounds of a number of other parameters restricted to the family of oriented triangle-free planar graphs using our second major contribution.

cs.DM↗

arXiv2022

On clique numbers of colored mixed graphs

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 itself to distinct image vertices under any homomorphism is called an (m,n)-relative clique of G. The maximum cardinality of an (m,n)-relative clique of a graph is called the (m,n)-relative clique number of the graph. In this article, we explore the (m,n)-relative clique numbers for various families of graphs.

math.CO↗

arXiv2020

On the signed chromatic number of some classes of graphs

A signed graph $(G, σ)$ is a graph $G$ along with a function $σ: E(G) \to \{+,-\}$. 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 and signs of closed walks. The signed chromatic number of a signed graph $(G, σ)$ is the minimum number of vertices $|V(H)|$ of a signed graph $(H, π)$ to which $(G, σ)$ admits a homomorphism.Homomorphisms of signed graphs have been attracting growing attention in the last decades, especially due to their strong connections to the theories of graph coloring and graph minors. These homomorphisms have been particularly studied through the scope of the signed chromatic number. In this work, we provide new results and bounds on the signed chromatic number of several families of signed graphs (planar graphs, triangle-free planar graphs, $K_n$-minor-free graphs, and bounded-degree graphs).

cs.DM↗

arXiv2019

Pushable chromatic number of graphs with degree constraints

Pushable homomorphisms and the pushable chromatic number $χ_p$ of oriented graphs were introduced by Klostermeyer and MacGillivray in 2004. They notably observed that, for any oriented graph $\overrightarrow{G}$, we have $χ_p(\overrightarrow{G}) \leq χ_o(\overrightarrow{G}) \leq 2 χ_p(\overrightarrow{G})$, where $χ_o(\overrightarrow{G})$ denotes the oriented chromatic number of $\overrightarrow{G}$. This stands as first general bounds on $χ_p$. This parameter was further studied in later works.This work is dedicated to the pushable chromatic number of oriented graphs fulfilling particular degree conditions. For all $Δ\geq 29$, we first prove that the maximum value of the pushable chromatic number of an oriented graph with maximum degree $Δ$ lies between $2^{\fracΔ{2}-1}$ and $(Δ-3) \cdot (Δ-1) \cdot 2^{Δ-1} + 2$ which implies an improved bound on the oriented chromatic number of the same family of graphs. For subcubic oriented graphs, that is, when $Δ\leq 3$, we then prove that the maximum value of the pushable chromatic number is~$6$ or~$7$. We also prove that the maximum value of the pushable chromatic number of oriented graphs with maximum average degree less than~$3$ lies between~$5$ and~$6$. The former upper bound of~$7$ also holds as an upper bound on the pushable chromatic number of planar oriented graphs with girth at least~$6$.

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arXiv2018

On relative clique number of colored mixed graphs

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 ($m, n)$-colored mixed chromatic number of an $(m, n)$-colored mixed graph $G$, introduced by Nešetřil and Raspaud [J. Combin. Theory Ser. B 2000] is the order (number of vertices) of the smallest homomorphic image of $G$. Later Bensmail, Duffy and Sen [Graphs Combin. 2017] introduced another parameter related to the $(m, n)$-colored mixed chromatic number, namely, the $(m, n)$-relative clique number as the maximum cardinality of a vertex subset which, pairwise, must have distinct images with respect to any colored homomorphism. In this article, we study the $(m, n$)-relative clique number for the family of subcubic graphs, graphs with maximum degree $Δ$, planar graphs and triangle-free planar graphs and provide new improved bounds in each of the cases. In particular, for subcubic graphs we provide exact value of the parameter.

cs.DM↗

arXiv2018

Erratum for "On oriented cliques with respect to push operation"

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 above list was missing an arc. We correct the error and restate the corrected statement in this article. We also point out the reason for the error and comment that the error occurred due to a mistake in a particular lemma. We present the corrected proof of that particular lemma as well. Moreover, a few counts were wrongly reported due to the above mentioned error. So we update our reported counts after correction in this article.

cs.DM↗

arXiv2016

Chromatic number of signed graphs with bounded maximum degree

A signed graph $ (G, Σ)$ is a graph positive and negative ($Σ$ denotes the set of negative edges). To re-sign a vertex $v$ of a signed graph $ (G, Σ)$ is to switch the signs of the edges incident to $v$. If one can obtain $ (G, Σ')$ by re-signing some vertices of $ (G, Σ)$, then $ (G, Σ) \equiv (G, Σ')$. A signed graphs $ (G, Σ)$ admits an homomorphism to $ (H, Λ)$ if there is a sign preserving vertex mapping from $(G,Σ')$ to $(H, Λ)$ for some $ (G, Σ) \equiv (G, Σ')$. The signed chromatic number $χ_{s}( (G, Σ))$ of the signed graph $(G, Σ)$ is the minimum order (number of vertices) of a signed graph $(H, Λ)$ such that $ (G, Σ)$ admits a homomorphism to $(H, Λ)$. For a family $ \mathcal{F}$ of signed graphs $χ_{s}(\mathcal{F}) = \text{max}_{(G,Σ) \in \mathcal{F}} χ_{s}( (G, Σ))$. We prove $2^{Δ/2-1} \leq χ_s(\mathcal{G}_Δ) \leq (Δ-1)^2. 2^{(Δ-1)} +2$ for all $Δ\geq 3$ where $\mathcal{G}_Δ$ is the family of connected signed graphs with maximum degree $Δ$. \end{abstract}

math.CO↗

arXiv2015

Approximation algorithms for the two-center problem of convex polygon

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 $r\leq2r_{opt}$ using $O(1)$ space, where $r_{opt}$ is the optimum solution. Next, we show that in non-streaming setup, we can improve the approximation factor by $r\leq 1.84 r_{opt}$, maintaining the time complexity of the algorithm to $O(n)$, and using $O(1)$ extra space in addition to the space required for storing the input.

cs.CG↗

arXiv2015

On oriented cliques with respect to push operation

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 remains an oriented clique even if one pushes any set of vertices of it. We show that it is NP-complete to decide if an undirected graph is underlying graph of a push clique or not. We also prove that a planar push clique can have at most 8 vertices. We also provide an exhaustive list of minimal (with respect to spanning subgraph inclusion) planar push cliques.

math.CO↗

arXiv2015

On chromatic number of colored mixed graphs

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 \textit{$(m,n)$-colored mixed chromatic number} $χ_{(m,n)}(G)$ of an $(m,n)$-colored mixed graph $G$ is the order (number of vertices) of the smallest homomorphic image of $G$. This notion was introduced by Nešetřil and Raspaud (2000, J. Combin. Theory, Ser. B 80, 147--155). They showed that $χ_{(m,n)}(G) \leq k(2m+n)^{k-1}$ where $G$ is a $k$-acyclic colorable graph. We proved the tightness of this bound. We also showed that the acyclic chromatic number of a graph is bounded by $k^2 + k^{2 + \lceil log_{(2m+n)} log_{(2m+n)} k \rceil}$ if its $(m,n)$-colored mixed chromatic number is at most $k$. Furthermore, using probabilistic method, we showed that for graphs with maximum degree $Δ$ its $(m,n)$-colored mixed chromatic number is at most $2(Δ-1)^{2m+n} (2m+n)^{Δ-1}$. In particular, the last result directly improves the upper bound $2Δ^2 2^Δ$ of oriented chromatic number of graphs with maximum degree $Δ$, obtained by Kostochka, Sopena and Zhu (1997, J. Graph Theory 24, 331--340) to $2(Δ-1)^2 2^{Δ-1}$. We also show that there exists a graph with maximum degree $Δ$ and $(m,n)$-colored mixed chromatic number at least $(2m+n)^{Δ/ 2}$.

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