Affirmative Solutions On Local Antimagic Chromatic Number

Lau, Gee-Choon; Ng, Ho-Kuen; Shiu, Wai Chee

doi:10.48550/arxiv.1805.02886

Cited by 5 publications

(6 citation statements)

References 4 publications

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“…(1) In [5,Theorem 5], the authors proved that every G = W 4k , k ≥ 1 with e = 8k edges admits a local antimagic labeling with χ la (W 4k ) = 3 such that for k ≥ 2, e < c 1 = 9k + 2 < c 2 = 11k + 1 < c 3 = 2k(12k + 1), while W 4 has c 1 = 11, c 2 = 15, c 3 = 20. Moreover, n 1 = n 2 = 2k, n 3 = 1.…”

Section: Adding Pendant Edgesmentioning

confidence: 99%

On number of pendants in local antimagic chromatic number

Lau¹,

Shiu²,

Ng³

2020

Preprint

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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, f + (x) = f + (y), where the induced vertex label f + (x) = f (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. Let χ(G) be the chromatic number of G. In this paper, sharp upper and lower bounds of χ la (G) for G with pendant vertices, and sufficient conditions for the bounds to equal, are obtained. Consequently, for k ≥ 1, there are infinitely many graphs with k ≥ χ(G) − 1 pendant vertices and χ la (G) = k + 1. We conjecture that every tree T k , other than certain caterpillars, spiders and lobsters, with k ≥ 1 pendant vertices has χ la (T k ) = k + 1.

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Section: Adding Pendant Edgesmentioning

confidence: 99%

On number of pendants in local antimagic chromatic number

Lau¹,

Shiu²,

Ng³

2020

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“…Note that K(2; 1, 0) ∼ = P 3 with χ la (P 3 ) = 3 and K(2; 1, 1) ∼ = P 4 with χ la (P 4 ) = 3 (see [1,Theorem 2.7]). Moreover, χ la (K(2; 2, 1)) = 4 (see [3,Theorem 8]). Observe that K(1; n − 1) ∼ = K(2; n − 2, 0) is the star graph K 1,n−1 of order n with χ la (K 1,n−1 ) = n.…”

Section: (M+n)mentioning

confidence: 99%

On local antimagic chromatic number of graphs with cut-vertices

Lau¹,

Shiu²,

Ng³

2018

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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, f + (x) = f + (y), where the induced vertex label f + (x) = f (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, the sharp lower bound of the local antimagic chromatic number of a graph with cut-vertices given by pendants is obtained. Conjecture 3.1 in [Affirmative solutions on local antimagic chromatic number (2018), submitted] is completely solved. The exact value of the local antimagic chromatic number of many families of graphs with cutvertices are also determined. Consequently, we partially answered Problem 3.1 in [Local antimagic vertex coloring of a graph, Graphs and Combin., 33 (2017) 275-285.].

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“…Thus, we have obtained various sufficient conditions such that χ la (K(p, q, r)) = 3. Interested readers may refer to [2,[5][6][7] for more results on local antimatic chromatic number of graphs.…”

Section: So We Havementioning

confidence: 99%

Cartesian Magicness of 3-Dimensional Boards

Lau¹,

Ng²,

Shiu³

2018

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A (p, q, r)-board that has pq + pr + qr squares consists of a (p, q)-, a (p, r)-, and a (q, r)-rectangle. Let S be the set of the squares. Consider a bijection f : S → [1, pq + pr + qr]. Firstly, for 1 ≤ i ≤ p, let xi be the sum of all the q + r integers in the i-th row of the (p, q + r)-rectangle. Secondly, for 1 ≤ j ≤ q, let yj be the sum of all the p + r integers in the j-th row of the (q, p + r)-rectangle. Finally, for 1 ≤ k ≤ r, let z k be the the sum of all the p + q integers in the k-th row of the (r, p + q)-rectangle. Such an assignment is called a (p, q, r)-design if {xi : 1 ≤ i ≤ p} = {c1} for some constant c1, {yj : 1 ≤ j ≤ q} = {c2} for some constant c2, and {z k : 1 ≤ k ≤ r} = {c3} for some constant c3. A (p, q, r)-board that admits a (p, q, r)-design is called (1) Cartesian tri-magic if c1, c2 and c3 are all distinct; (2) Cartesian bi-magic if c1, c2 and c3 assume exactly 2 distinct values; (3) Cartesian magic if c1 = c2 = c3 (which is equivalent to supermagic labeling of K(p, q, r)). Thus, Cartesian magicness is a generalization of magic rectangles into 3-dimensional space. In this paper, we study the Cartesian magicness of various (p, q, r)-board by matrix approach involving magic squares or rectangles. In Section 2, we obtained various sufficient conditions for (p, q, r)-boards to admit a Cartesian tri-magic design. In Sections 3 and 4, we obtained many necessary and (or) sufficient conditions for various (p, q, r)-boards to admit (or not admit) a Cartesian bi-magic and magic design. In particular, it is known that K(p, p, p) is supermagic and thus every (p, p, p)-board is Cartesian magic. We gave a short and simpler proof that every (p, p, p)-board is Cartesian magic.

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Affirmative Solutions On Local Antimagic Chromatic Number

Cited by 5 publications

References 4 publications

On number of pendants in local antimagic chromatic number

On number of pendants in local antimagic chromatic number

On local antimagic chromatic number of graphs with cut-vertices

Cartesian Magicness of 3-Dimensional Boards

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