Nearly magic rectangles

Chai, Feng-Shun; Singh, Rakhi; Stufken, John

doi:10.1002/jcd.21667

Cited by 10 publications

(5 citation statements)

References 10 publications

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“…). Define g : E(G) → [1, 4mn + 2m + n(2n − 3) − 1] such that g(e) = h(e) if e ∈ E(C 2m−1 ), g(v i u j ) = x i,j + (2m − 1) for 1 ≤ i ≤ 2n, 1 ≤ j ≤ 2m − 1 where x i,j is the (i, j)-entry of a (2n, 2m−1)-nearly magic rectangle that has (2i−1)-st row sum (2m−1)(1+4mn−2n)−1 2 = n(2m−1) 2 +m−1, (2i)-th row sum (2m−1)(1+4mn−2n)+1 2 = n(2m−1) 2 +m and column sum n(1+4mn−2n) (see [6]). Finally, g(e) = f (e) + (2n + 1)(2m − 1) if e ∈ K 2n .…”

Section: Resultsmentioning

confidence: 99%

On local antimagic chromatic number of cycle-related join graphs

Lau¹,

Shiu²,

Ng³

2021

Discuss. Math. Graph Theory

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An edge labeling of a connected graph G = (V, E) is said to be local antimagic if it is a bijection f : E → {1, . . . , |E|} such that for any pair of adjacent vertices x and y, ff (e), with e ranging over all the edges incident to x. The local antimagic chromatic number of G, denoted by χ la (G), is the minimum number of distinct induced vertex labels over all local antimagic labelings of G. In this paper, several sufficient conditions for χ la (H) ≤ χ la (G) are obtained, where H is obtained from G with a certain edge deleted or added. We then determined the exact value of the local antimagic chromatic number of many cycle related join graphs.

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Section: Resultsmentioning

confidence: 99%

On local antimagic chromatic number of cycle-related join graphs

Lau¹,

Shiu²,

Ng³

2021

Discuss. Math. Graph Theory

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“…In [2], the authors generalized the concept of magic rectangle to nearly magic rectangle as follows.…”

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confidence: 99%

On join product and local antimagic chromatic number of regular graphs

Lau¹,

Shiu²

2022

Preprint

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Let G = (V, E) be a connected simple graph of order p and size q. A graph G is called local antimagic if G admits a local antimagic labeling. A bijection f : E → {1, 2, . . . , q} is called a local antimagic labeling of G if for any two adjacent vertices u and v, we have f, and E(u) is the set of edges incident to u. Thus, any local antimagic labeling induces a proper vertex coloring of G if vertex v is assigned the color f + (v). The local antimagic chromatic number, denoted χ la (G), is the minimum number of induced colors taken over local antimagic labeling of G. Let G and H be two vertex disjoint graphs. The join graph of G andIn this paper, we show the existence of non-complete regular graphs with arbitrarily large order, regularity and local antimagic chromatic numbers.

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“…In this article, we prove the existence of QMR a b d ( , : ) for all possible values of a and b, when d ab = 2 + 1 ∕ . In addition, if gcd a b ( , ) = 1, we prove that the condition A magic square is an n n × array whose entries are the consecutive numbers n 1, 2, …, 2 , each appearing once, such that the sum of each row, column, and the main and main backward diagonal is equal to n n ( +1) 2…”

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confidence: 87%

“…When a and b are of different parity, a magic rectangle MR a b ( , ) does not exist. Recently, Chai et al [2] introduced the concept of nearly magic rectangles. A nearly magic rectangle NMR a b ( , ) is defined as an a b × array with a odd and b even that contains each of the integers from the set ab {1, 2, …, } exactly once and the row sums are constant, while the column sums differ by at most one.…”

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confidence: 99%