Symmetry

Research

14 pages, 329 KiB

Open AccessArticle

Gregarious Decompositions of Complete Equipartite Graphs and Related Structures

by Zsolt Tuza

Symmetry 2023, 15(12), 2097; https://doi.org/10.3390/sym15122097 - 22 Nov 2023

Viewed by 1028

In a finite mathematical structure with a given partition, a substructure is said to be gregarious if either it meets each partition class or it shares at most one element with each partition class. In this paper, we considered edge decompositions of graphs [...] Read more.

In a finite mathematical structure with a given partition, a substructure is said to be gregarious if either it meets each partition class or it shares at most one element with each partition class. In this paper, we considered edge decompositions of graphs and hypergraphs into gregarious subgraphs and subhypergraphs. We mostly dealt with “complete equipartite” graphs and hypergraphs, where the vertex classes have the same size and precisely those edges or hyperedges of a fixed cardinality are present that do not contain more than one element from any class. Some related graph classes generated by product operations were also considered. The generalization to hypergraphs offers a wide open area for further research. Full article

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

11 pages, 471 KiB

Open AccessArticle

Edge Resolvability in Generalized Petersen Graphs

by Tanveer Iqbal, Syed Ahtsham Ul Haq Bokhary, Shreefa O. Hilali, Mohammed Alhagyan, Ameni Gargouri and Muhammad Naeem Azhar

Symmetry 2023, 15(9), 1633; https://doi.org/10.3390/sym15091633 - 24 Aug 2023

Viewed by 933

Abstract

The generalized Petersen graphs are a type of cubic graph formed by connecting the vertices of a regular polygon to the corresponding vertices of a star polygon. This graph has many interesting graph properties. As a result, it has been widely researched. In [...] Read more.

The generalized Petersen graphs are a type of cubic graph formed by connecting the vertices of a regular polygon to the corresponding vertices of a star polygon. This graph has many interesting graph properties. As a result, it has been widely researched. In this work, the edge metric dimensions of the generalized Petersen graphs GP(2l + 1, l) and GP(2l, l) are explored, and it is shown that the edge metric dimension of GP(2l + 1, l) is equal to its metric dimension. Furthermore, it is proved that the upper bound of the edge metric dimension is the same as the value of the metric dimension for the graph GP(2l, l). Full article

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

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30 pages, 363 KiB

Open AccessFeature PaperArticle

Minimal Non-C-Perfect Hypergraphs with Circular Symmetry

by Péter Bence Czaun, Pál Pusztai, Levente Sebők and Zsolt Tuza

Symmetry 2023, 15(5), 1114; https://doi.org/10.3390/sym15051114 - 19 May 2023

Viewed by 4348

Abstract

In this research paper, we study 3-uniform hypergraphs

H = (X, E)

with circular symmetry. Two parameters are considered: the largest size

α (H)

of a set

S \subset X

not containing any edge

E \in E

, [...] Read more.

In this research paper, we study 3-uniform hypergraphs

H = (X, E)

with circular symmetry. Two parameters are considered: the largest size

α (H)

of a set

S \subset X

not containing any edge

E \in E

, and the maximum number

\bar{χ} (H)

of colors in a vertex coloring of

H

such that each

E \in E

contains two vertices of the same color. The problem considered here is to characterize those

H

in which the equality

\bar{χ} (H^{'}) = α (H^{'})

holds for every induced subhypergraph

H^{'} = (X^{'}, E^{'})

of

H

. A well-known objection against

\bar{χ} (H^{'}) = α (H^{'})

is where

|\cap_{E \in E^{'}} E| = 1

, termed “monostar”. Steps toward a solution to this approach is to investigate the properties of monostar-free structures. All such

H

are completely identified up to 16 vertices, with the aid of a computer. Most of them can be shown to satisfy

\bar{χ} (H) = α (H)

, and the few exceptions contain one or both of two specific induced subhypergraphs

H_{5}^{^{⥁}}

,

H_{6}^{^{⥁}}

on five and six vertices, respectively, both with

\bar{χ} = 2

and

α = 3

. Furthermore, a general conjecture is raised for hypergraphs of prime orders. Full article

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

17 pages, 887 KiB

Open AccessArticle

Metric-Based Fractional Dimension of Rotationally-Symmetric Line Networks

by Rashad Ismail, Muhammad Javaid and Hassan Zafar

Symmetry 2023, 15(5), 1069; https://doi.org/10.3390/sym15051069 - 12 May 2023

Viewed by 1716

Abstract

The parameter of distance plays an important role in studying the properties symmetric networks such as connectedness, diameter, vertex centrality and complexity. Particularly different metric-based fractional models are used in diverse fields of computer science such as integer programming, pattern recognition, and in [...] Read more.

The parameter of distance plays an important role in studying the properties symmetric networks such as connectedness, diameter, vertex centrality and complexity. Particularly different metric-based fractional models are used in diverse fields of computer science such as integer programming, pattern recognition, and in robot navigation. In this manuscript, we have computed all the local resolving neighborhood sets and established sharp bounds of a metric-based fractional dimension called by the local fractional metric dimension of the rotationally symmetric line networks of wheel and prism networks. Furthermore, the bounded and unboundedness of these networks is also checked under local fractional metric dimension when the order of these networks approaches to infinity. The lower and upper bounds of local fractional metric dimension of all the rotationally symmetric line networks is also analyzed by using

3 D

shapes. Full article

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

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9 pages, 242 KiB

Open AccessArticle

Resolvability in Subdivision Graph of Circulant Graphs

by Syed Ahtsham Ul Haq Bokhary, Khola Wahid, Usman Ali, Shreefa O. Hilali, Mohammed Alhagyan and Ameni Gargouri

Symmetry 2023, 15(4), 867; https://doi.org/10.3390/sym15040867 - 5 Apr 2023

Cited by 1 | Viewed by 1545

Abstract

Circulant networks are a very important and widely studied class of graphs due to their interesting and diverse applications in networking, facility location problems, and their symmetric properties. The structure of the graph ensures that it is symmetric about any line that cuts [...] Read more.

Circulant networks are a very important and widely studied class of graphs due to their interesting and diverse applications in networking, facility location problems, and their symmetric properties. The structure of the graph ensures that it is symmetric about any line that cuts the graph into two equal parts. Due to this symmetric behavior, the resolvability of these graph becomes interning. Subdividing an edge means inserting a new vertex on the edge that divides it into two edges. The subdivision graph G is a graph formed by a series of edge subdivisions. In a graph, a resolving set is a set that uniquely identifies each vertex of the graph by its distance from the other vertices. A metric basis is a resolving set of minimum cardinality, and the number of elements in the metric basis is referred to as the metric dimension. This paper determines the minimum resolving set for the graphs

H_{l} [1, k]

constructed from the circulant graph

C_{l} [1, k]

by subdividing its edges. We also proved that, for

k = 2, 3

, this graph class has a constant metric dimension. Full article

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

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26 pages, 713 KiB

Open AccessArticle

On Rotationally Symmetrical Planar Networks and Their Local Fractional Metric Dimension

by Shahbaz Ali, Rashad Ismail, Francis Joseph H. Campena, Hanen Karamti and Muhammad Usman Ghani

Symmetry 2023, 15(2), 530; https://doi.org/10.3390/sym15020530 - 16 Feb 2023

Cited by 4 | Viewed by 1800

Abstract

The metric dimension has various applications in several fields, such as computer science, image processing, pattern recognition, integer programming problems, drug discovery, and the production of various chemical compounds. The lowest number of vertices in a set with the condition that any vertex [...] Read more.

The metric dimension has various applications in several fields, such as computer science, image processing, pattern recognition, integer programming problems, drug discovery, and the production of various chemical compounds. The lowest number of vertices in a set with the condition that any vertex can be uniquely identified by the list of distances from other vertices in the set is the metric dimension of a graph. A resolving function of the graph G is a map

ϑ : V (G) \to [0, 1]

such that

\sum_{u \in R {v, w}} ϑ (u) \geq 1,

for every pair of adjacent distinct vertices

v, w \in V (G)

. The local fractional metric dimension of the graph G is defined as

{ldim}_{f} (G)

=

min {\sum_{v \in V (G)} ϑ (v)

, where

ϑ

is a local resolving function of

G}

. This paper presents a new family of planar networks namely, rotationally heptagonal symmetrical graphs by means of up to four cords in the heptagonal structure, and then find their upper-bound sequences for the local fractional metric dimension. Moreover, the comparison of the upper-bound sequence for the local fractional metric dimension is elaborated both numerically and graphically. Furthermore, the asymptotic behavior of the investigated sequences for the local fractional metric dimension is addressed. Full article

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

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14 pages, 800 KiB

Open AccessArticle

The Ascending Ramsey Index of a Graph

by Gary Chartrand and Ping Zhang

Symmetry 2023, 15(2), 523; https://doi.org/10.3390/sym15020523 - 15 Feb 2023

Cited by 2 | Viewed by 1495

Abstract

Let G be a graph with a given red-blue coloring c of the edges of G. An ascending Ramsey sequence in G with respect to c is a sequence

G_{1}

,

G_{2}

, …,

G_{k}

of pairwise edge-disjoint [...] Read more.

Let G be a graph with a given red-blue coloring c of the edges of G. An ascending Ramsey sequence in G with respect to c is a sequence

G_{1}

,

G_{2}

, …,

G_{k}

of pairwise edge-disjoint subgraphs of G such that each subgraph

G_{i}

(

1 \leq i \leq k

) is monochromatic and

G_{i}

is isomorphic to a proper subgraph of

G_{i + 1}

(

1 \leq i \leq k - 1

). The ascending Ramsey index

A R_{c} (G)

of G with respect to c is the maximum length of an ascending Ramsey sequence in G with respect to c. The ascending Ramsey index

A R (G)

of G is the minimum value of

A R_{c} (G)

among all red-blue colorings c of G. It is shown that there is a connection between this concept and set partitions. The ascending Ramsey index is investigated for some classes of highly symmetric graphs such as complete graphs, matchings, stars, graphs consisting of a matching and a star, and certain double stars. Full article

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

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15 pages, 427 KiB

Open AccessArticle

On the P₃ Coloring of Graphs

by Hong Yang, Muhammad Naeem and Shahid Qaisar

Symmetry 2023, 15(2), 521; https://doi.org/10.3390/sym15020521 - 15 Feb 2023

Cited by 1 | Viewed by 2259

Abstract

The vertex coloring of graphs is a well-known coloring of graphs. In this coloring, all of the vertices are assigned colors in such a way that no two adjacent vertices have the same color. We can call this type of coloring

P_{2}

[...] Read more.

The vertex coloring of graphs is a well-known coloring of graphs. In this coloring, all of the vertices are assigned colors in such a way that no two adjacent vertices have the same color. We can call this type of coloring

P_{2}

coloring, where

P_{2}

is a path graph. However, there are situations in which this type of coloring cannot give us the solution to the problem at hand. To answer such questions, in this article, we introduce a novel graph coloring called

P_{3}

coloring. A graph is called

P_{3}

-colorable if we can assign colors to the vertices of the graph such that the vertices of every

P_{3}

path are distinct. The minimum number of colors required for a graph to have

P_{3}

coloring is called the

P_{3}

chromatic number. The aim of this article is, in general, to prove some basic results concerning this coloring, and, in particular, to compute the

P_{3}

chromatic number for different symmetric families of graphs. Full article

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

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12 pages, 361 KiB

Open AccessArticle

Modular Version of Edge Irregularity Strength for Fan and Wheel Graphs

by Debi Oktia Haryeni, Zata Yumni Awanis, Martin Bača and Andrea Semaničová-Feňovčíková

Symmetry 2022, 14(12), 2671; https://doi.org/10.3390/sym14122671 - 16 Dec 2022

Cited by 3 | Viewed by 1860

Abstract

A k-labeling from the vertex set of a simple graph

G = (V, E)

to a set of integers

{1, 2, \dots, k}

is defined to be a modular edge irregular if, for every [...] Read more.

A k-labeling from the vertex set of a simple graph

G = (V, E)

to a set of integers

{1, 2, \dots, k}

is defined to be a modular edge irregular if, for every couple of distinct edges, their modular edge weights are distinct. The modular edge weight is the remainder of the division of the sum of end vertex labels by modulo

| E (G) |

. The modular edge irregularity strength of a graph is known as the maximal vertex label k, minimized over all modular edge irregular k-labelings of the graph. In this paper we describe labeling schemes with symmetrical distribution of even and odd edge weights and investigate the existence of (modular) edge irregular labelings of joins of paths and cycles with isolated vertices. We estimate the bounds of the (modular) edge irregularity strength for the join graphs

P_{n} + \bar{K_{m}}

and

C_{n} + \bar{K_{m}}

and determine the corresponding exact value of the (modular) edge irregularity strength for some fan graphs and wheel graphs in order to prove the sharpness of the presented bounds. Full article

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

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18 pages, 480 KiB

Open AccessArticle

Generalized Arithmetic Staircase Graphs and Their Total Edge Irregularity Strengths

by Yeni Susanti, Sri Wahyuni, Aluysius Sutjijana, Sutopo Sutopo and Iwan Ernanto

Symmetry 2022, 14(9), 1853; https://doi.org/10.3390/sym14091853 - 6 Sep 2022

Cited by 3 | Viewed by 1673

Abstract

Let

Γ = (V_{Γ}, E_{Γ})

be a simple undirected graph with finite vertex set

V_{Γ}

and edge set

E_{Γ}

. A total

n

-labeling [...] Read more.

Let

Γ = (V_{Γ}, E_{Γ})

be a simple undirected graph with finite vertex set

V_{Γ}

and edge set

E_{Γ}

. A total

n

-labeling

α : V_{Γ} \cup E_{Γ} \to {1, 2, \dots, n}

is called a total edge irregular labeling on

Γ

if for any two different edges

x y

and

x^{'} y^{'}

in

E_{Γ}

the numbers

α (x) + α (x y) + α (y)

and

α (x^{'}) + α (x^{'} y^{'}) + α (y^{'})

are distinct. The smallest positive integer n such that

Γ

can be labeled by a total edge irregular labeling is called the total edge irregularity strength of the graph

Γ

. In this paper, we provide the total edge irregularity strength of some asymmetric graphs and some symmetric graphs, namely generalized arithmetic staircase graphs and generalized double-staircase graphs, as the generalized forms of some existing staircase graphs. Moreover, we give the construction of the corresponding total edge irregular labelings. Full article

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

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15 pages, 473 KiB

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 3 | Viewed by 1744

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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