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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 3188

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 1037

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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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 3779

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 3387

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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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 3742

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 4016

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

{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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