Properly even harmonious labelings of disconnected graphs

Gallian, Joseph A.; Stewart, Danielle

doi:10.1016/j.akcej.2015.11.015

Cited by 8 publications

(13 citation statements)

References 2 publications

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“…Theorem 2.3. (Gallian and Schoenhard [3]) kP 2 is properly even harmonious if and only if k is even.…”

Section: Preliminariesmentioning

confidence: 99%

Properly even harmonious labeling of a union of stars

Henderson

2024

EJGTA

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A function f is defined as an even harmonious labeling on a graph G with q edges if f : V (G) → {0, 1, . . . , 2q} is an injection and the induced function) is bijective. A properly even harmonious labeling is an even harmonious labeling in which the codomain of f is {0, 1, . . . , 2q − 1}, and a strongly harmonious labeling is an even harmonious labeling that also satisfies the additional condition that for any two adjacent vertices with labels u and v, 0 < u + v ≤ 2q. In [3], Gallian and Schoenhard proved thatIn this paper, we begin with the related question "When is the graph of k n-star components, G = kS n , properly even harmonious?" We conclude that kS n is properly even harmonious if and only if k is even or k is odd, k > 1, and n ≥ 2. We also conclude that S n 1 ∪ S n 2 ∪ • • • ∪ S n k is properly even harmonious when k ≥ 2, n i ≥ 2 for all i and give some additional results on combinations of star and banana graphs.

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“…Theorem 2.3. (Gallian and Schoenhard [3]) kP 2 is properly even harmonious if and only if k is even.…”

Section: Preliminariesmentioning

confidence: 99%

Properly even harmonious labeling of a union of stars

Henderson

2024

EJGTA

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“…The graph 𝐺 is an even harmonious if there is a one-to-one function f:V(G)→{0,1,2,… ,2q} so that each edge gets a different even label in the set of integers modulo 2q as the result of the sum of the two labels of the vertices incident to that edge. Even harmonious graphs that have been studied can be seen in [11], [12], [13], [14], [15].…”

Section: Theorem 1 (Liang and Baimentioning

confidence: 99%

The Harmonious, Odd Harmonious, and Even Harmonious Labeling

Lasim

Halikin

Wijaya

2022

BAREKENG: J. Math. & App.

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Suppose is a simple and connected graph with edges. A harmonious labeling on a graph is an injective function so that there exists a bijective function where for each An odd harmonious labeling on a graph is an injective function from to non-negative integer set less than so that there is a function where for every An even harmonious labeling on a graph is an injective function so that there is a bijective function where for each . In this paper, we discuss how to build new labeling (harmonious, odd harmonious, even harmonious) based on the existing labeling (harmonious, odd harmonious, even harmonious)

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“…In fact the constraints (2) and (3) make a bijection from V to the labels 1 2 V . Binary requirements on the x v i variables are given by (4).…”

Section: Modeling the Prime Labeling Of Graphsmentioning

confidence: 99%

“…In the intervening years dozens of graph labelings techniques have been studied in over 800 papers. 4 The prime labeling problem is one of the branches of this issue. A graph G with vertex set V is said to have a prime labeling if its vertices can be labeled with distinct integers 1 2 V such that for every edge xy in E, the labels assigned to x and y are relatively prime.…”

Section: Introductionmentioning

confidence: 99%

On the Prime Labeling of Graphs

Deng¹,

Du²

2014

Jnl of Comp & Theo Nano

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A graph G with vertex set V and edge set E is said to have a prime labeling if its vertices can be labeled with distinct integers 1 2 V such that for every edge xy in E, the labels assigned to x and y are relatively prime. A graph is called prime if it has a prime labeling. In this paper, we introduce to consider the extremal prime labeling graphs with the maximum number of edges. In addition, by modelling the prime labeling problem, we obtain some prime graphs in the families of Cartesian and direct product of paths and cycles.

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Properly even harmonious labelings of disconnected graphs

Cited by 8 publications

References 2 publications

Properly even harmonious labeling of a union of stars

Properly even harmonious labeling of a union of stars

The Harmonious, Odd Harmonious, and Even Harmonious Labeling

On the Prime Labeling of Graphs

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