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Open AccessFeature PaperArticle

On Vertex Magic 3-Regular Graphs with a Perfect Matching

by Tao-Ming Wang

Mathematics 2025, 13(24), 3969; https://doi.org/10.3390/math13243969 - 12 Dec 2025

Viewed by 1415

Let

G = (V, E)

be a finite simple graph with

p = | V |

vertices and

q = | E |

edges, without isolated vertices or isolated edges. A vertex magic total labeling is a bijection f from [...] Read more.

Let

G = (V, E)

be a finite simple graph with

p = | V |

vertices and

q = | E |

edges, without isolated vertices or isolated edges. A vertex magic total labeling is a bijection f from

V \cup E

to the consecutive integers

1, 2, \dots, p + q,

with the property that, for every vertex

u \in V

, one has

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

for some magic constant k. The vertex magic total labeling is called E-super if furthermore

f (E) = {1, 2, \dots, q}

. A graph is called (E-super) vertex magic if it admits an (E-super) vertex magic total labeling. In this paper, we verify the existence of E-super vertex magic total labeling for a class of 3-regular graphs with a perfect matching, and we confirm the existence of such a labeling for general regular graphs of odd degree containing particular classes of 3-factors, which provides us with known and new examples. Note that Harary graphs are among the popular models used in communication networks. In 2012, G. Marimuthu and M. Balakrishnan raised a conjecture that if

n > 4

n \equiv 0 (m o d 4)

and m is odd, then the Harary graph

H_{m, n}

admits an E-super vertex magic labeling. Among others, we are able to verify this conjecture except for one case while

m = 3

and

n \equiv 4 (m o d 8)

. Full article

(This article belongs to the Special Issue Graph Theory and Applications, 3rd Edition)

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17 pages, 1318 KB

Open AccessArticle

On Forbidden Subgraphs of (K₂, H)-Sim-(Super)Magic Graphs

by Yeva Fadhilah Ashari, A.N.M. Salman and Rinovia Simanjuntak

Symmetry 2021, 13(8), 1346; https://doi.org/10.3390/sym13081346 - 26 Jul 2021

Cited by 3 | Viewed by 3031

Abstract

A graph G admits an H-covering if every edge of G belongs to a subgraph isomorphic to a given graph H. G is said to be H-magic if there exists a bijection [...] Read more.

A graph G admits an H-covering if every edge of G belongs to a subgraph isomorphic to a given graph H. G is said to be H-magic if there exists a bijection

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

such that

w_{f} (H^{'}) = \sum_{v \in V (H^{'})} f (v) + \sum_{e \in E (H^{'})} f (e)

is a constant, for every subgraph

H^{'}

isomorphic to H. In particular, G is said to be H-supermagic if

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

. When H is isomorphic to a complete graph

K_{2}

, an H-(super)magic labeling is an edge-(super)magic labeling. Suppose that G admits an F-covering and H-covering for two given graphs F and H. We define G to be

(F, H)

-sim-(super)magic if there exists a bijection

f^{'}

that is simultaneously F-(super)magic and H-(super)magic. In this paper, we consider

(K_{2}, H)

-sim-(super)magic where H is isomorphic to three classes of graphs with varied symmetry: a cycle which is symmetric (both vertex-transitive and edge-transitive), a star which is edge-transitive but not vertex-transitive, and a path which is neither vertex-transitive nor edge-transitive. We discover forbidden subgraphs for the existence of

(K_{2}, H)

-sim-(super)magic graphs and classify classes of

(K_{2}, H)

-sim-(super)magic graphs. We also derive sufficient conditions for edge-(super)magic graphs to be

(K_{2}, H)

-sim-(super)magic and utilize such conditions to characterize some

(K_{2}, H)

-sim-(super)magic graphs. Full article

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

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