arXiv subjects

Aparna Lakshmanan S

Publications and source records attributed to Aparna Lakshmanan S.

18 recordsLinked to original sources

arXiv2026

Ramsey sequences with bounded clique size

A sequence of graphs $ \{G_k\} $ is a Ramsey sequence if for every positive integer $ k $, the graph $ G_k $ is a proper subgraph of $ G_{k+1} $, and there exists an integer $n > k$ such that every red-blue edge coloring of $ G_n $ contains a monochromatic copy of $ G_k $. Among the wide range of open problems in Ramsey theory, an interesting open question is ``Does there exist an ascending sequence $\{G_k\}$ with $\lim_{k \to \infty} χ(G_k) = \infty$ and $\lim_{k \to \infty} ω(G_k) \neq \infty$ that is a Ramsey sequence?". In this paper, we solve this problem by demonstrating that the sequence of General shift graphs $Sh(n,k,l)$ is a Ramsey sequence which satisfies the conditions given in the open problem. We further present an alternative proof of the generalised Ramsey theorem by establishing explicitly that $Sh(n,2)$ is a Ramsey sequence for $k$-edge-colorings, and by observing the natural bijection between the set of all $t$-element subsets of $[n]$ and the edge set $E(Sh(n,t-1))$.

math.CO↗

arXiv2026

Mutual-visibility Coloring of Graphs

The mutual-visibility chromatic number of a graph $G$ is the smallest number of colors needed to color the vertices of $G$ such that each color class is a mutual-visibility set. In this paper, we prove that determining the mutual-visibility chromatic number of a graph is NP-complete even when restricted to the class of graphs having diameter four and mutual-visibility chromatic number two. We further determine the exact value of the mutual-visibility chromatic number for glued binary trees and glued $t$-ary trees.

math.CO↗

arXiv2026

On a conjecture about the strong odd chromatic number of planar graphs

A proper coloring of a graph $G$ is said to be a strong odd coloring of $G$, if for every vertex $v$ and every color $c$, either $c$ appears on an odd number of vertices in the neighborhood of $v$ or $c$ is absent in the neighborhood of $v$. The strong odd chromatic number of $G$ is defined as the smallest integer $k$ for which $G$ admits a strong odd coloring using $k$ colors. In this paper, we evaluate the strong odd chromatic number of join of cycles and empty graphs and one point union of graphs. Using these results, we construct infinite family of planar graphs that serves as counter examples to a recent conjecture regarding the upper bound of the strong odd chromatic number of planar graphs.

math.CO↗

arXiv2025

Endomorphism and Automorphism Graphs

Let $G$ be a group. The directed endomorphism graph, \dend of $G$ is a directed graph with vertex set $G$ and there is a directed edge from the vertex `$a$' to the vertex `$\, b$' $(a \neq b) $ if and only if there exists an endomorphism on $G$ mapping $a$ to $b$. The endomorphism graph, \uend $\,$ of $G$ is the corresponding undirected simple graph. The automorphism graph, ${Auto}(G)$ of $G$ is an undirected graph with vertex set $G$ and there is an edge from the vertex `$a$' to the vertex `$\,b$' $(a \neq b) $ if and only if there exists an automorphism on $G$ mapping $a$ to $b$. We have explored graph theoretic properties like size, planarity, girth etc. and tried finding out for which types of groups these graphs are complete, diconnected, trees, bipartite and so on.

math.CO↗

arXiv2025

Endomorphism and automorphism graphs of finite groups

Let $G$ be a group. The directed endomorphism graph, $\dend(G)$ of $G$ is a directed graph with vertex set $G$ and there is a directed edge from the vertex $a$ to the vertex $b$ if $a \neq b$ and there exists an endomorphism on $G$ mapping $a$ to $b$. The endomorphism graph, $\uend(G)$ is the corresponding undirected simple graph. The automorphism graph of $G$ is similarly defined for automorphisms: it is a disjoint union of complete graphs on the orbits of $\Aut(G)$. The endomorphism digraph is a special case of a digraph associated with a transformation monoid, and we begin by introducing this. We have explored graph theoretic properties like size, planarity, girth etc. and tried finding out for which types of groups these graphs are complete, diconnected, trees, bipartite and so on, as well as computing these graphs for some special groups. We conclude with examples showing that things are not always simple.

math.GR↗

arXiv2025

The vertex visibility number of graphs

If $x\in V(G)$, then $S\subseteq V(G)\setminus\{x\}$ is an $x$-visibility set if for any $y\in S$ there exists a shortest $x,y$-path avoiding $S$. The $x$-visibility number $v_x(G)$ is the maximum cardinality of an $x$-visibility set, and the maximum value of $v_x(G)$ among all vertices $x$ of $G$ is the vertex visibility number ${\rm vv}(G)$ of $G$. It is proved that ${\rm vv}(G)$ is equal to the largest possible number of leaves of a shortest-path tree of $G$. Deciding whether $v_x(G) \ge k$ holds for given $G$, a vertex $x\in V(G)$, and a positive integer $k$ is NP-complete even for graphs of diameter $2$. Several general sharp lower and upper bounds on the vertex visibility number are proved. The vertex visibility number of Cartesian products is also bounded from below and above, and the exact value of the vertex visibility number is determined for square grids, square prisms, and square toruses.

cs.DM↗

arXiv2025

Distance mutual-visibility coloring: relations with (total) domination, exact distance graphs and graph products

The concept of mutual-visibility (MV) has been extended in several directions. A vertex subset $S$ of a graph $G$ is a $k$-distance mutual-visibility ($k$DMV) set if for any two vertices in $S$, there is a geodesic between them of length at most $k$ whose internal vertices are not in $S$. In this paper, we combine this with the MV coloring as follows. For any integer $k\geq1$, a $k$DMV coloring of $G$ is a partition of $V(G)$ into $k$DMV sets, and the $k$DMV chromatic number $χ_{μ_k}(G)$ is the minimum cardinality of such a partition. When $k=1$ or $k\ge {\rm diam}(G)$, it equals the clique cover number $θ(G)$ or the MV chromatic number $χ_μ(G)$, respectively. So, our attention is given to $1<k<{\rm diam}(G)$ with $k=2$ producing the most interesting results. We prove that $χ_{μ_2}(G)\le|V(G)|/2$ and present large families of graphs that attain the bound. In addition, $χ_{μ_2}(G)$ is bounded from above by the total domination number $γ_t(G)$ if $G$ is isolate-free, while in graphs $G$ with girth $g(G)\geq7$, $χ_{μ_2}(G)$ is bounded from below by the domination number $γ(G)$. A surprising relation with the exact distance-2 graphs is found, which results in $θ(G^{[\natural2]})=γ_{t}(G)$ for any isolate-free graph $G$ with $g(G)\geq7$. The relation is explored further in lexicographic product graphs, where we prove the sharp inequalities $χ_{μ_{2}}(G\circ H)\leq θ(G^{[\natural2]})\leq θ\big{(}(G\circ H)^{[\natural2]}\big{)}$. We also prove a sharp lower (resp. upper) bound on $χ_{μ_2}$ (resp. $χ_{μ_k}$) for the Cartesian (resp. strong) product of two connected graphs and show that they are widely sharp. Finally, we characterize the block graphs $G$ with $χ_{μ_k}(G)=χ_μ(G)$, where $k={\rm diam}(G)-1$.

math.CO↗

arXiv2025

The k-Sudoku Number of Graphs

Let $G=(V,E)$ be a graph of order $n$ with chromatic number $χ(G)$. Let $ k \geq χ(G) $ and $S \subseteq V$. Let $ C_0 $ be a $k$-coloring of the induced subgraph $ G[S] $. The coloring $C_0$ is called an extendable coloring, if $C_0$ can be extended to a $k$-coloring of $G$ and it is a $k$- Sudoku coloring of $G$, if $C_0$ can be uniquely extended to a $k$-coloring of $G$. The smallest order of such an induced subgraph $G[S]$ of $G$ which admits a $k$- Sudoku coloring is called $k$- Sudoku number of $G$ and is denoted by $sn(G,k)$. When $k=χ(G)$, we call $k$- Sudoku number of $G$ as Sudoku number of $G$ and is denoted by $sn(G)$. In this paper, we have obtained the $3$- Sudoku number of some bipartite graphs $P_n$, $C_{2n}$, $K_{m,n}$, $B_{m,n}$ and $G \circ lK_1$, where $G$ is a bipartite graph and $l\geq1$. Also, we have obtained the necessary and sufficient conditions for a bipartite graph $G$ to have $sn(G,3)$ equal to $n$, $n-1$ or $n-2$. Also, we study the relation between $k$- Sudoku number of a graph $G$ and the Sudoku number of a supergraph $H$ of $G$.

math.CO↗

arXiv2025

Erdős Conjecture and AR-Labeling

Given an edge labeling $f$ of a graph $G$, a vertex $v$ is called an $AR$-vertex, if $v$ has distinct edge weight sums for each distinct subset of edges incident on $v$. An injective edge labeling $f$ of a graph $G$ is called an $AR$-labeling of $G$, if $f:E(G) \rightarrow \mathbb{N}$ is such that every vertex in $G$ is an $AR$-vertex under $f$. The minimum $k$ such that there exists an $AR$-labeling $f:E\rightarrow \{1,2,3,\dots,k\}$ is called the $AR$-index of G, denoted by $ARI(G)$. In this paper, using a sequence originating from Erdős subset sum conjecture, a lower bound has been obtained for the $AR$-index of a graph and this bound is used to prove that only finitely many bistars, complete graphs and complete bipartite graphs are $AR$-graphs. The exact values of $AR$-index is obtained for stars and wheels.

math.CO↗

arXiv2025

On Geodesic Leech Labeling of Some Graph Classes

Let $f:E\rightarrow \{1,2,3,\dots\}$ be an edge labeling of $G$. The geodesic path number of $G$, $t_{gp}(G)$, is the number of geodesic paths in $G$. An edge labeling $f$ is called a geodesic Leech labeling, if the set of weights of the geodesic paths in $G$ is $\{1,2,3,\dots,t_{gp}(G)\}$, where the weight of a path $P$ is the sum of the labels assigned to the edges of $P$. A graph which admits a geodesic Leech labeling is called a geodesic Leech graph. Otherwise, we call it a non-geodesic Leech graph. In this paper, we prove that cycles $C_n$, $n \geq 5$ are non-geodesic Leech graphs. We also prove that there are at most three regular complete bipartite graphs that are geodesic Leech. We show that degree sequence cannot characterize geodesic Leech graphs. The geodesic path number of the wheel graph $W_n$ is obtained and the geodesic Leech labeling of $W_5$ and $W_6$ is given.

math.CO↗

arXiv2025

Sudoku Number of Corona of Graphs

Let $G = (V,E)$ be a graph of order $n$ with chromatic number $χ(G) = k$, let $S \subset V$ and let $C_0$ be a $k$-coloring of the induced subgraph $G[S]$. The coloring $C_0$ is called an extendable coloring, if $C_0$ can be extended to a $k$-coloring of $G$ and it is a Sudoku coloring of $G$ if the extension is unique. The smallest order of such an induced subgraph $G[S]$ of $G$ which admits a Sudoku coloring is called the Sudoku number of $G$ and is denoted by $sn(G)$. In this paper, we first introduce the notion of uniquely color extendable vertex and then we obtain the lower and upper bounds for the Sudoku number of $G \circ K_1$. Some families of graphs which attain these bounds are also obtained. The exact value of the Sudoku number of corona of $C_n$, $W_n$ and $K_n$ with $K_1$ and $C_n \circ P_m$ are also obtained.

math.CO↗

arXiv2025

An edge labeling of graphs from Rados partition regularity condition

A vertex $v$ is called an AR-vertex, if $v$ has distinct edge weight sums for each distinct subset of edges incident on $v$. i.e., if $\{x_1,x_2,\dots,x_k\}$ are the edge labels of the edges incident on $v$, then the $2^k$ subset sums are all distinct. An injective edge labeling $f$ of a graph $G$ is said to be an AR-labeling of $G$, if $f:E \rightarrow \mathbb{N}$ is such that every vertex in $G$ is an AR-vertex under $f$. A graph $G$ is said to be an AR-graph, if there exists an AR-labeling $f:E\rightarrow \{1,2,\dots,m\}$, where $m$ denotes the number of edges of $G$. A study of AR-labeling and AR-graphs is initiated in this paper.

math.CO↗

arXiv2024

Counting largest mutual-visibility and general position sets of glued $t$-ary trees

All four invariants of the mutual-visibility problem and, all four invariants of the general position problem are determined for glued binary trees. The number of the corresponding extremal sets is obtained in each of the eight situations. The results are further extended to glued $t$-ary trees, and some of them also to generalized glued binary trees.

math.CO↗

arXiv2022

Impact of Vertex Addition on Italian Domination Number

An Italian dominating function (IDF), of a graph G is a function $ f: V(G) \rightarrow \{0,1,2\} $ satisfying the condition that for every $ v\in V(G) $ with $ f(v) = 0, \sum_{ u\in N(v)} f(u) \geq 2. $ The weight of an IDF on $G$ is the sum $ f(V)= \sum_{v\in V(G)}f(v) $ and Italian domination number, $ γ_I(G) $ is the minimum weight of an IDF. In this paper, we study the impact of corona operator and addition of twins on Italian domination number.

math.CO↗

arXiv2021

Perfect Italian Domination Number of Graphs

In this paper, an upper bound for the perfect Italian domination number of the cartesian product of any two graphs is obtained and the exact value of this parameter for cartesian product of some special graphs are obtained. We have also proved that for any two positive integers $a$, $b$ there exists a graph $G$ and an induced subgraph $H$ of $G$ such that $γ_I^p(G) = a$ and $γ_I^p(H) = b$. Relationship of the perfect Italian domination number with the Roman domination number and the perfect domination number of a graph $G$ are obtained and the corresponding realization problems are also solved. We have also obtained the perfect Italian domination number of the Mycielskian of a graph in terms of the perfect domination number of the graph. Some open problems related to this parameters are also included.

math.CO↗

arXiv2021

The transformation of edge-regular and pseudo strongly regular graphs under graph operations

The graph $G$ is said to be strongly regular with parameters $(n,k,λ,μ)$ if the following conditions hold: (1) each vertex has $k$ neighbours; (2) any two adjacent vertices of $G$ have $λ$ common neighbours; (3) any two non-adjacent vertices of $G$ have $μ$ common neighbours. In this paper we study two weaker notions of strongly regular graphs. A graph satisfying the conditions $(1)$ and $(2)$ is called an edge-regular graph with parameters $(n,k,λ)$. We call a graph satisfying the conditions $(1)$ and $(3)$ a pseudo strongly regular graph with parameters $(n,k,μ)$. In this paper we study the impact of various graph operations on edge regular graphs and pseudo strongly regular graphs.

math.CO↗

arXiv2021

Spectrum of Strongly Regular Graphs under Graph Operators

In this paper, we show that if G is strongly regular then the Gallai graph and the anti-Gallai graph of G are edge-regular. We also identify conditions under which the Gallai and anti-Gallai graphs are themselves strongly regular, as well as conditions under which they are 2-connected. We include also a number of concrete examples and a discussion of spectral properties of the Gallai and anti-Gallai graphs.

math.CO↗

arXiv2020

Italian Domination and Perfect Italian Domination on Sierpinski Graphs

An Italian dominating function (IDF) of a graph G is a function $ f: V(G) \rightarrow \{0,1,2\} $ satisfying the condition that for every $ v\in V $ with $ f(v) = 0$, $\sum_{ u\in N(v)} f(u) \geq 2. $ The weight of an IDF on $G$ is the sum $ f(V)= \sum_{v\in V}f(v) $ and the Italian domination number, $ γ_I(G) $, is the minimum weight of an IDF. An IDF is a perfect Italian dominating function (PID) on $G$, if for every vertex $ v \in V(G) $ with $ f(v) = 0 $ the total weight assigned by $f$ to the neighbours of $ v $ is exactly 2, i.e., all the neighbours of $u$ are assigned the weight 0 by $f$ except for exactly one vertex $v$ for which $ f(v) = 2 $ or for exactly two vertices $v$ and $w$ for which $ f(v) = f(w) = 1 $. The weight of a PID- function is $f(V)=\sum_{u \in V(G)}f(u)$. The perfect Italian domination number of $G$, denoted by $ γ^{p}_{I}(G), $ is the minimum weight of a PID-function of $G$. In this paper we obtain the Italian domination number and perfect Italian domination number of Sierpiński graphs.

math.CO↗