A magical approach to some labeling conjectures

Figueroa-Centeno, Ramón M.; Ichishima, Rikio; Muntaner-Batle, Francesc A.; Oshima, Atsushi

doi:10.7151/dmgt.1531

Cited by 20 publications

(12 citation statements)

References 13 publications

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“…Therefore, it seems to be a natural question to ask which 2-regular graphs are super edge-magic. This is what was in the minds of Figueroa-Centeno et al in [10] when they conjectured that the graph C m ∪ C n is super edge-magic if and only if m + n is odd and greater than 1. Holden et al went further into this conjecture, although they arrived to it from a different point of view, when they conjectured in [12] that all 2-regular graphs of odd order are strong vertex total magic, excluding C 3 ∪ C 4 , 3C 3 ∪ C 4 and 2C 3 ∪ C 5 , which is in fact equivalent to saying that they are super edge-magic.…”

Section: Super Edge-magic Labelings Of 2-regular Graphsmentioning

confidence: 65%

The Jumping Knight and Other (Super) Edge-Magic Constructions

2013

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Let G be a graph of order p and size q with loops allowed. A bijective function f : V (G) ∪ E(G) → {i} p+q i=1 is an edge-magic labeling of G if the sum f (u) + f (uv) + f (v) = k is independent of the choice of the edge uv. The constant k is called either the valence, the magic weight or the magic sum of the labeling f. If a graph admits an edge-magic labeling, then it is called an edge-magic graph. Furthermore, if the function f meets the extra condition that f (V (G)) = {i} p i=1 then f is called a super edge-magic labeling and G is called a super edge-magic graph. A digraph D admits a labeling, namely l, if its underlying graph, und(D) admits l. In this paper, we introduce a new construction of super edge-magic labelings which is related to the classical jump of the knight on the chess game. We also use super edge-magic labelings of digraphs together with a generalization of the Kronecker product in order to get edge-magic labelings of some families of graphs.

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Section: Super Edge-magic Labelings Of 2-regular Graphsmentioning

confidence: 65%

The Jumping Knight and Other (Super) Edge-Magic Constructions

2013

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“…Motivated by Conjecture 3.2, Figueroa-Centeno et al [17] showed that some 2-regular graphs with two components are SEM. They proved that 3 ∪…”

Section: Conjecture 32 [16] a 2-regular Graph Of Odd Order Is Sem Imentioning

confidence: 99%

“…Motivated by Conjecture 3.2 , Figueroa-Centeno et al [17] showed that some 2-regular graphs with two components are SEM. They proved that

is SEM if and only if

4

is even,

is SEM if and only if

5

is odd,

is SEM if and only if

4

is even, and

is SEM for any even

4

and odd

4

.…”

Section: The Semd Of 2 and Related Graphsmentioning

confidence: 99%

On the super edge-magic deficiency of some graphs

Krisnawati

Ngurah

Hidayat

et al. 2020

Heliyon

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Forest with two components 2-regular graph Join product graph A graph is called super edge-magic if there exists a bijection ∶ () ∪ () ⟶ {1, 2, ⋯ , | ()| + | ()|}, where (()) = {1, 2, ⋯ , | ()|}, such that () + () + () is a constant for every edge ∈ (). Such a case, is called a super edge magic labeling of. A bipartite graph with partite sets and is called consecutively super edge-magic if there exists a super edge-magic labeling with the property that () = {1, 2, ⋯ , | |} and () = {| | + 1, | | + 2, ⋯ , | ()|}. The super edge-magic deficiency of a graph , denoted by (), is either the minimum nonnegative integer such that ∪ 1 is super edge-magic or +∞ if there exists no such. The consecutively super edge-magic deficiency of a bipartite graph , denoted by (), is either the minimum nonnegative integer such that ∪ 1 is consecutively super edge-magic or +∞ if there exists no such. In this paper, we study the super edge-magic deficiency of some graphs. We investigate the (consecutively) super edge-magic deficiency of forests with two components. We also investigate the super edge-magic deficiency of a 2-regular graph 2 3 ∪ and join product of 1, ∪ with an isolated vertex.

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“…Based on the results of Theorem 1, the super edge-magic deficiency of H n is 0 for n = 3 and 4, and at least 1 for n ≥ 5. For n = 5, 6, 7, we could prove that µ s (H n ) = 1 by labeling the vertices (c; 6 ), and (c; x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 ) with (1; 7, 5, 3, 6, 4), (2; 3, 1, 4, 8, 5, 6), and (2; 3, 1,4,8,5,9,6), respectively.…”

Section: Super Edge-magic Deficiency Of a Wheel Minus An Edgementioning

confidence: 99%