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Open AccessArticle

Graceful Local Antimagic Labeling of Graphs: A Pattern Analysis Using Python

by Luqman Alam, Andrea Semaničová-Feňovčíková and Ioan-Lucian Popa

Symmetry 2025, 17(1), 108; https://doi.org/10.3390/sym17010108 - 12 Jan 2025

Cited by 1 | Viewed by 2350

Graph labeling is the process of assigning labels to vertices and edges under certain conditions. This paper investigates the graceful local antimagic labeling of various graph families, excluding symmetric labelings, using computational experiments and Python-based algorithms. Through these experiments, we identify new results [...] Read more.

Graph labeling is the process of assigning labels to vertices and edges under certain conditions. This paper investigates the graceful local antimagic labeling of various graph families, excluding symmetric labelings, using computational experiments and Python-based algorithms. Through these experiments, we identify new results and patterns within specific graph classes. The study expands on the existing literature by offering computational evidence, proposing algorithms for the verification of labelings, and exploring the relationship between the local antimagic labeling and the chromatic number. Our results increase the understanding of graph labeling and offer insights into its computational aspects. Full article

(This article belongs to the Special Issue Symmetry and Graph Theory, 2nd Edition)

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17 pages, 307 KB

Open AccessArticle

On Bridge Graphs with Local Antimagic Chromatic Number 3

by Wai-Chee Shiu, Gee-Choon Lau and Ruixue Zhang

Mathematics 2025, 13(1), 16; https://doi.org/10.3390/math13010016 - 25 Dec 2024

Viewed by 801

Abstract

Let

G = (V, E)

be a connected graph. A bijection

f : E \to {1, \dots, | E |}

is called a local antimagic labeling if, for any two adjacent vertices x and y, [...] Read more.

Let

G = (V, E)

be a connected graph. A bijection

f : E \to {1, \dots, | E |}

is called a local antimagic labeling if, for any two adjacent vertices x and y,

f^{+} (x) \neq f^{+} (y)

, where

f^{+} (x) = \sum_{e \in E (x)} f (e)

, and

E (x)

is the set of edges incident to x. Thus, a local antimagic labeling induces a proper vertex coloring of G, where the vertex x is assigned the color

f^{+} (x)

. The local antimagic chromatic number

χ_{l a} (G)

is the minimum number of colors taken over all colorings induced by local antimagic labelings of G. In this paper, we present some families of bridge graphs with

χ_{l a} (G) = 3

and give several ways to construct bridge graphs with

χ_{l a} (G) = 3

. Full article

(This article belongs to the Special Issue Advances in Graph Theory: Algorithms and Applications)

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9 pages, 552 KB

Open AccessArticle

A Study of Independency on Fuzzy Resolving Sets of Labelling Graphs

by Ramachandramoorthi Shanmugapriya, Perichetla Kandaswamy Hemalatha, Lenka Cepova and Jiri Struz

Mathematics 2023, 11(16), 3440; https://doi.org/10.3390/math11163440 - 8 Aug 2023

Cited by 4 | Viewed by 1939

Abstract

Considering a fuzzy graph G is simple and can be connected and considered as a subset [...] Read more.

Considering a fuzzy graph G is simple and can be connected and considered as a subset

H = \{(u_{1}, σ (u_{1})), (u_{2}, σ (u_{2})), \dots (u_{k}, σ (u_{k}))\}, | H | \geq 2

; then, every two pairs of elements of

σ - H

have a unique depiction with the relation of

H

, and

H

can be termed as a fuzzy resolving set (FRS). The minimal

H

cardinality is regarded as the fuzzy resolving number (FRN), and it is signified by

F r (G)

. An independence set is discussed on the FRS, fuzzy resolving domination set (FRDS), and Fuzzy modified antimagic resolving set (FMARS). In this paper, we discuss the independency of FRS and FMARS in which an application has also been developed. Full article

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10 pages, 284 KB

Open AccessArticle

On the Study of Rainbow Antimagic Connection Number of Comb Product of Friendship Graph and Tree

by Brian Juned Septory, Liliek Susilowati, Dafik Dafik and Veerabhadraiah Lokesha

Symmetry 2023, 15(1), 12; https://doi.org/10.3390/sym15010012 - 21 Dec 2022

Cited by 4 | Viewed by 2177

Abstract

Given a graph G with vertex set

V (G)

and edge set

E (G)

, for the bijective function [...] Read more.

Given a graph G with vertex set

V (G)

and edge set

E (G)

, for the bijective function

f (V (G)) \to {1, 2, \dots, | V (G) |}

, the associated weight of an edge

x y \in E (G)

under f is

w (x y) = f (x) + f (y)

. If all edges have pairwise distinct weights, the function f is called an edge-antimagic vertex labeling. A path P in the vertex-labeled graph G is said to be a rainbow

x - y

path if for every two edges

x y, x^{'} y^{'} \in E (P)

it satisfies

w (x y) \neq w (x^{'} y^{'})

. The function f is called a rainbow antimagic labeling of G if there exists a rainbow

x - y

path for every two vertices

x, y \in V (G)

. We say that graph G admits a rainbow antimagic coloring when we assign each edge

x y

with the color of the edge weight

w (x y)

. The smallest number of colors induced from all edge weights of antimagic labeling is the rainbow antimagic connection number of G, denoted by

r a c (G)

. This paper is intended to investigate non-symmetrical phenomena in the comb product of graphs by considering antimagic labeling and optimizing rainbow connection, called rainbow antimagic coloring. In this paper, we show the exact value of the rainbow antimagic connection number of the comb product of graph

F_{n} ⊳ T_{m}

, where

F_{n}

is a friendship graph with order

2 n + 1

and

T_{m} \in {P_{m}, S_{m}, B r_{m, p}, S_{m, m}}

, where

P_{m}

is the path graph of order m,

S_{m}

is the star graph of order

m + 1

,

B r_{m, p}

is the broom graph of order

m + p

and

S_{m, m}

is the double star graph of order

2 m + 2

. Full article

(This article belongs to the Section Mathematics)

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15 pages, 473 KB

Open AccessArticle

Distance Antimagic Product Graphs

by Rinovia Simanjuntak and Aholiab Tritama

Symmetry 2022, 14(7), 1411; https://doi.org/10.3390/sym14071411 - 9 Jul 2022

Cited by 5 | Viewed by 2577

Abstract

A distance antimagic graph is a graph G admitting a bijection

f : V (G) \to {1, 2, \dots, | V (G) |}

such that for two distinct vertices x and y, [...] Read more.

A distance antimagic graph is a graph G admitting a bijection

f : V (G) \to {1, 2, \dots, | V (G) |}

such that for two distinct vertices x and y,

ω (x) \neq ω (y)

, where

ω (x) = \sum_{y \in N (x)} f (y)

, for

N (x)

the open neighborhood of x. It was conjectured that a graph G is distance antimagic if and only if G contains no two vertices with the same open neighborhood. In this paper, we study several distance antimagic product graphs. The products under consideration are the three fundamental graph products (Cartesian, strong, direct), the lexicographic product, and the corona product. We investigate the consequence of the non-commutative (or sometimes called non-symmetric) property of the last two products to the antimagicness of the product graphs. Full article

(This article belongs to the Special Issue Labelings, Colorings and Distances in Graphs)

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12 pages, 289 KB

Open AccessArticle

Antimagic Labeling for Product of Regular Graphs

by Vinothkumar Latchoumanane and Murugan Varadhan

Symmetry 2022, 14(6), 1235; https://doi.org/10.3390/sym14061235 - 14 Jun 2022

Cited by 5 | Viewed by 7150

Abstract

An antimagic labeling of a graph

G = (V, E)

is a bijection from the set of edges of G to

\{1, 2, \dots, |E (G)|\}

and such that any two vertices of G have [...] Read more.

An antimagic labeling of a graph

G = (V, E)

is a bijection from the set of edges of G to

\{1, 2, \dots, |E (G)|\}

and such that any two vertices of G have distinct vertex sums where the vertex sum of a vertex v in

V (G)

is nothing but the sum of all the incident edge labeling of G. In this paper, we discussed the antimagicness of rooted product and corona product of graphs. We proved that if we let G be a connected t-regular graph and H be a connected k-regular graph, then the rooted product of graph G and H admits antimagic labeling if

t \geq k

. Moreover, we proved that if we let G be a connected t-regular graph and H be a connected k-regular graph, then the corona product of graph G and H admits antimagic labeling for all

t, k \geq 2

. Full article

(This article belongs to the Special Issue Graph Labelings and Their Applications)

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9 pages, 482 KB

Open AccessArticle

Linear Forest mP₃ Plus a Longer Path P_n Becoming Antimagic

by Jen-Ling Shang and Fei-Huang Chang

Mathematics 2022, 10(12), 2036; https://doi.org/10.3390/math10122036 - 12 Jun 2022

Viewed by 1682

Abstract

An edge labeling of a graph G is a bijection f from

E (G)

to a set of

| E (G) |

integers. For a vertex u in G, the induced vertex sum of u, denoted by [...] Read more.

An edge labeling of a graph G is a bijection f from

E (G)

to a set of

| E (G) |

integers. For a vertex u in G, the induced vertex sum of u, denoted by

f^{+} (u)

, is defined as

f^{+} (u) = \sum_{u v \in E (G)} f (u v)

. Graph G is said to be antimagic if it has an edge labeling g such that

g (E (G)) = {1, 2, \dots, | E (G) |}

and

g^{+} (u) \neq g^{+} (v)

for any pair

u, v \in V (G)

with

u \neq v

. A linear forest is a union of disjoint paths of orders greater than one. Let

m P_{k}

denote a linear forest consisting of m disjoint copies of path

P_{k}

. It is known that

m P_{3}

is antimagic if and only if

m = 1

. In this study, we add a disjoint path

P_{n}

(

n \geq 4

) to

m P_{3}

and develop a necessary condition and a sufficient condition whereby the new linear forest

m P_{3} ⋃ P_{n}

may be antimagic. Full article

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18 pages, 369 KB

Open AccessArticle

The Local Antimagic Total Chromatic Number of Some Wheel-Related Graphs

by Xue Yang, Hong Bian, Haizheng Yu and Dandan Liu

Axioms 2022, 11(3), 97; https://doi.org/10.3390/axioms11030097 - 25 Feb 2022

Cited by 1 | Viewed by 3490

Abstract

Let

G = (V, E)

be a connected graph with

| V | = n

and

| E | = m

. A bijection [...] Read more.

Let

G = (V, E)

be a connected graph with

| V | = n

and

| E | = m

. A bijection

f : V (G) \cup E (G) \to {1, 2, \dots, n + m}

is called local antimagic total labeling if, for any two adjacent vertices u and v,

ω_{t} (u) \neq ω_{t} (v)

, where

ω_{t} (u) = f (u) + \sum_{e \in E (u)} f (e)

, and

E (u)

is the set of edges incident to u. Thus, any local antimagic total labeling induces a proper coloring of G, where the vertex x in G is assigned the color

ω_{t} (x)

. The local antimagic total chromatic number, denoted by

χ_{l a t} (G)

, is the minimum number of colors taken over all colorings induced by local antimagic total labelings of G. In this paper, we present the local antimagic total chromatic numbers of some wheel-related graphs, such as the fan graph

F_{n}

, the bowknot graph

B_{n, n}

, the Dutch windmill graph

D_{4}^{n}

, the analogous Dutch graph

A D_{4}^{n}

and the flower graph

F_{n}

. Full article

(This article belongs to the Special Issue Graph Theory with Applications)

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13 pages, 307 KB

Open AccessArticle

The Local Antimagic Chromatic Numbers of Some Join Graphs

by Xue Yang, Hong Bian, Haizheng Yu and Dandan Liu

Math. Comput. Appl. 2021, 26(4), 80; https://doi.org/10.3390/mca26040080 - 22 Nov 2021

Cited by 7 | Viewed by 3089

Abstract

Let

G = (V (G), E (G))

be a connected graph with n vertices and m edges. A bijection

f : E (G) \to {1, 2, \dots, m}

is [...] Read more.

Let

G = (V (G), E (G))

be a connected graph with n vertices and m edges. A bijection

f : E (G) \to {1, 2, \dots, m}

is an edge labeling of G. For any vertex x of G, we define

ω (x) = \sum_{e \in E (x)} f (e)

as the vertex label or weight of x, where

E (x)

is the set of edges incident to x, and f is called a local antimagic labeling of G, if

ω (u) \neq ω (v)

for any two adjacent vertices

u, v \in V (G)

. It is clear that any local antimagic labelling of G induces a proper vertex coloring of G by assigning the vertex label

ω (x)

to any vertex x of G. The local antimagic chromatic number of G, denoted by

χ_{l a} (G)

, is the minimum number of different vertex labels taken over all colorings induced by local antimagic labelings of G. In this paper, we present explicit local antimagic chromatic numbers of

F_{n} \lor \bar{K_{2}}

and

F_{n} - v

, where

F_{n}

is the friendship graph with n triangles and v is any vertex of

F_{n}

. Moreover, we explicitly construct an infinite class of connected graphs G such that

χ_{l a} (G) = χ_{l a} (G \lor \bar{K_{2}})

, where

G \lor \bar{K_{2}}

is the join graph of G and the complement graph of complete graph

K_{2}

. This fact leads to a counterexample to a theorem of Arumugam et al. in 2017, and our result also provides a partial solution to Problem 3.19 in Lau et al. in 2021. Full article

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15 pages, 683 KB

Open AccessArticle

Another Antimagic Conjecture

by Rinovia Simanjuntak, Tamaro Nadeak, Fuad Yasin, Kristiana Wijaya, Nurdin Hinding and Kiki Ariyanti Sugeng

Symmetry 2021, 13(11), 2071; https://doi.org/10.3390/sym13112071 - 2 Nov 2021

Cited by 5 | Viewed by 2936

Abstract

An antimagic labeling of a graph G is a bijection

f : E (G) \to {1, \dots, | E (G) |}

such that the weights [...] Read more.

An antimagic labeling of a graph G is a bijection

f : E (G) \to {1, \dots, | E (G) |}

such that the weights

w (x) = \sum_{y \sim x} f (y)

distinguish all vertices. A well-known conjecture of Hartsfield and Ringel (1990) is that every connected graph other than

K_{2}

admits an antimagic labeling. For a set of distances D, a D-antimagic labeling of a graph G is a bijection

f : V (G) \to {1, \dots, | V (G) |}

such that the weight

ω (x) = \sum_{y \in N_{D} (x)} f (y)

is distinct for each vertex x, where

N_{D} (x) = {y \in V (G) | d (x, y) \in D}

is the D-neigbourhood set of a vertex x. If

N_{D} (x) = r

, for every vertex x in G, a graph G is said to be $(D, r)$ -regular. In this paper, we conjecture that a graph admits a D-antimagic labeling if and only if it does not contain two vertices having the same D-neighborhood set. We also provide evidence that the conjecture is true. We present computational results that, for

D = {1}

, all graphs of order up to 8 concur with the conjecture. We prove that the set of

(D, r)

-regular D-antimagic graphs is closed under union. We provide examples of disjoint union of symmetric

(D, r)

-regular that are D-antimagic and examples of disjoint union of non-symmetric non-

(D, r)

-regular graphs that are D-antimagic. Furthermore, lastly, we show that it is possible to obtain a D-antimagic graph from a previously known distance antimagic graph. Full article

(This article belongs to the Special Issue Graph Labelings and Their Applications)

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12 pages, 247 KB

Open AccessArticle

Local Antimagic Chromatic Number for Copies of Graphs

by Martin Bača, Andrea Semaničová-Feňovčíková and Tao-Ming Wang

Mathematics 2021, 9(11), 1230; https://doi.org/10.3390/math9111230 - 27 May 2021

Cited by 18 | Viewed by 3438

Abstract

An edge labeling of a graph

G = (V, E)

using every label from the set

{1, 2, \dots, | E (G) |}

exactly once is a local antimagic labeling if the vertex-weights [...] Read more.

An edge labeling of a graph

G = (V, E)

using every label from the set

{1, 2, \dots, | E (G) |}

exactly once is a local antimagic labeling if the vertex-weights are distinct for every pair of neighboring vertices, where a vertex-weight is the sum of labels of all edges incident with that vertex. Any local antimagic labeling induces a proper vertex coloring of G where the color of a vertex is its vertex-weight. This naturally leads to the concept of a local antimagic chromatic number. The local antimagic chromatic number is defined to be the minimum number of colors taken over all colorings of G induced by local antimagic labelings of G. In this paper, we estimate the bounds of the local antimagic chromatic number for disjoint union of multiple copies of a graph. Full article

(This article belongs to the Special Issue Advances in Discrete Applied Mathematics and Graph Theory)

17 pages, 296 KB

Open AccessArticle

Local Super Antimagic Total Labeling for Vertex Coloring of Graphs

by Slamin Slamin, Nelly Oktavia Adiwijaya, Muhammad Ali Hasan, Dafik Dafik and Kristiana Wijaya

Symmetry 2020, 12(11), 1843; https://doi.org/10.3390/sym12111843 - 7 Nov 2020

Cited by 13 | Viewed by 3725

Abstract

Let

G = (V, E)

be a graph with vertex set V and edge set E. A local antimagic total vertex coloring f of a graph G with vertex-set V and edge-set E is an injective map from [...] Read more.

Let

G = (V, E)

be a graph with vertex set V and edge set E. A local antimagic total vertex coloring f of a graph G with vertex-set V and edge-set E is an injective map from

V \cup E

to

{1, 2, \dots, | V | + | E |}

such that if for each

u v \in E (G)

then

w (u) \neq w (v)

, where

w (u) = \sum_{u v \in E (G)} f (u v) + f (u)

. If the range set f satisfies

f (V) = {1, 2, \dots, | V |}

, then the labeling is said to be local super antimagic total labeling. This labeling generates a proper vertex coloring of the graph G with the color

w (v)

assigning the vertex v. The local super antimagic total chromatic number of graph G,

χ_{l s a t} (G)

is defined as the least number of colors that are used for all colorings generated by the local super antimagic total labeling of G. In this paper we investigate the existence of the local super antimagic total chromatic number for some particular classes of graphs such as a tree, path, cycle, helm, wheel, gear, sun, and regular graphs as well as an amalgamation of stars and an amalgamation of wheels. Full article

(This article belongs to the Special Issue Graph Labelings and Their Applications)

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