Face antimagic labelings of toroidal and Klein bottle grid graphs

Butt, Saad Ihsan; Numan, Muhammad; Ali, Sharafat; Semaničová-Feňovčíková, Andrea

doi:10.1016/j.akcej.2018.09.005

Cited by 3 publications

(2 citation statements)

References 6 publications

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“…Wang [14] showed that the toroidal grid graphs C m × C n are antimagic for all integers m, n ≥ 3. Butt et al [5] investigated face antimagic labelings on toroidal and Klein bottle grid graphs. Curran, Low and Locke [6,7] investigated C 4 -face-magic toroidal labelings on C m × C n and C 4 -face-magic cylindrical labelings on P m × C n .…”

Section: Introductionmentioning

confidence: 99%

Equatorially balanced C4-face-magic labelings on Klein bottle grid graphs

Curran¹,

Low²,

Locke³

2022

Preprint

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For a graph G = (V, E) embedded in the Klein bottle, let F(G) denote the set of faces of G. Then, G is called a C k -face-magic Klein bottle graph if there exists a bijection f :We consider the m × n grid graph, denoted by Km,n, embedded in the Klein bottle in the natural way. We show that for m, n ≥ 2, Km,n admits a C 4 -face-magic Klein bottle labeling if and only if n is even. We say that a C 4 -face-magic Klein bottle labeling {x i,j : (i, j) ∈ V (Km,n)} on Km,n is equatorially balanced if x i,j + x i,n+1−j = 1 2 S for all (i, j) ∈ V (Km,n). We show that when m is odd, a C 4 -face-magic Klein bottle labeling on Km,n must be equatorially balanced. Also when m is odd, we show that (up to symmetries on the Klein bottle) the number of C 4 -face-magic Klein bottle labelings on the m × 4 Klein bottle grid graph is 2 m (m − 1)! τ (m), where τ (m) is the number of positive divisors of m.Furthermore, let m ≥ 3 be an odd integer and n ≥ 6 be an even integer. Then, the minimum number of distinct C 4 -face-magic Klein bottle labelings X on Km,n (up to symmetries on a Klein bottle) is either (5 • 2 m )(m − 1)! if n ≡ 0 (mod 4), or (6 • 2 m )(m − 1)! if n ≡ 2 (mod 4).

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Section: Introductionmentioning

confidence: 99%

Equatorially balanced C4-face-magic labelings on Klein bottle grid graphs

Curran¹,

Low²,

Locke³

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Wang [13] showed that the toroidal grid graphs C m × C n are antimagic for all integers m, n ≥ 3. Butt et al [6] investigated face antimagic labelings on toroidal and Klein bottle grid graphs.…”

Section: Introductionmentioning

confidence: 99%

C_4-face-magic toroidal labelings on C_m × C_n

Curran

Low

Locke

2020

Art Discrete Appl. Math.

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For a graph G = (V, E) naturally embedded in the torus, let F(G) denote the set of faces of G. Then, G is called a C n -face-magic toroidal graph if there exists a bijection f : V (G) → {1, 2, . . . , |V (G)|} such that for every F ∈ F(G) with F ∼ = C n , the sum of all the vertex labels along C n is a constant S. LetWe show that, for all m, n ≥ 2, C m × C n admits a C 4 -face-magic toroidal labeling if and only if either m = 2, or n = 2, or both m and n are even. We say that a C 4 -face-magic toroidal labeling {x i,j :We show that there exists an antipodal balanced C 4 -facemagic toroidal labeling on C 2m × C 2n if and only if the parity of m and n are the same. Furthermore, when both m and n are even, an antipodal balanced C 4 -face-magic toroidal labeling on C 2m × C 2n is both row-sum balanced and column-sum balanced. In addition, when m = n is even, an antipodal balanced C 4 -face-magic toroidal labeling on C 2n × C 2n is diagonal-sum balanced.

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$$C_4$$-Face-Magic Labelings on Even Order Projective Grid Graphs

Curran,

Locke

2024

Springer Proceedings in Mathematics &Amp; Statistics

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Face antimagic labelings of toroidal and Klein bottle grid graphs

Cited by 3 publications

References 6 publications

Equatorially balanced C4-face-magic labelings on Klein bottle grid graphs

Equatorially balanced C4-face-magic labelings on Klein bottle grid graphs

C_4-face-magic toroidal labelings on C_m × C_n

$$C_4$$-Face-Magic Labelings on Even Order Projective Grid Graphs

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