Distance Antimagic Labelings of Graphs

Kamatchi, Nainarraj; Vijayakumar, G. R.; Ramalakshmi, A.; Nilavarasi, S.; Arumugam, S.

doi:10.1007/978-3-319-64419-6_15

Cited by 8 publications

(19 citation statements)

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“…Corollary 1 is a generalization of a result in [15], where it was proved that if G is a regular distance antimagic graph, then 2G is also distance antimagic.…”

Section: End If 12: End Functionmentioning

confidence: 79%

“…Another type of antimagic labeling was introduced by Kamatchi and Arumugam in 2013 [14]. A bijection f : V(G) → {1, 2, ..., v} is called a distance antimagic labeling of graph G if for two distinct vertices x and y, w(x) = ω(y), where…”

Section: Introductionmentioning

confidence: 99%

“…In the same paper, Kamatchi and Arumugam conjectured that a graph G is distance antimagic if and only if G does not have two vertices with the same open neighbourhood. Some families of graphs have been shown to be distance antimagic, among others, the path P n , the cycle C n (n = 4), the wheel W n (n = 4) [14], and the hypercube Q n (n ≥ 3) [15]. In 2016, Llado and Miller [16] utilized Combinatorial Nullstellensatz to prove that a tree with l leaves and 2l vertices is distance antimagic.…”

Section: Introductionmentioning

confidence: 99%

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Another Antimagic Conjecture

et al. 2021

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An antimagic labeling of a graph G is a bijection f:E(G)→{1,…,|E(G)|} such that the weights w(x)=∑y∼xf(y) distinguish all vertices. A well-known conjecture of Hartsfield and Ringel (1990) is that every connected graph other than K2 admits an antimagic labeling. For a set of distances D, a D-antimagic labeling of a graph G is a bijection f:V(G)→{1,…,|V(G)|} such that the weightω(x)=∑y∈ND(x)f(y) is distinct for each vertex x, where ND(x)={y∈V(G)|d(x,y)∈D} is the D-neigbourhood set of a vertex x. If ND(x)=r, for every vertex x in G, a graph G is said to be (D,r)-regular. In this paper, we conjecture that a graph admits a D-antimagic labeling if and only if it does not contain two vertices having the same D-neighborhood set. We also provide evidence that the conjecture is true. We present computational results that, for D={1}, all graphs of order up to 8 concur with the conjecture. We prove that the set of (D,r)-regular D-antimagic graphs is closed under union. We provide examples of disjoint union of symmetric (D,r)-regular that are D-antimagic and examples of disjoint union of non-symmetric non-(D,r)-regular graphs that are D-antimagic. Furthermore, lastly, we show that it is possible to obtain a D-antimagic graph from a previously known distance antimagic graph.

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“…Corollary 1 is a generalization of a result in [15], where it was proved that if G is a regular distance antimagic graph, then 2G is also distance antimagic.…”

Section: End If 12: End Functionmentioning

confidence: 79%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Another Antimagic Conjecture

et al. 2021

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“…, n} such that at any vertex x, the weight of x, ω(x) = ∑ y∈N(x) f (y) is constant, where N(x) is the open neighborhood of x, i.e., the set of vertices adjacent to x. In 2013, the notion of distance antimagic labeling of a graph G was then introduced by Kamatchi and Arumugam [2]. A bijection f : V(G) → {1, 2, .…”

Section: Introductionmentioning

confidence: 99%

Distance Antimagic Product Graphs

Simanjuntak

Tritama

2022

Symmetry

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A distance antimagic graph is a graph G admitting a bijection f:V(G)→{1,2,…,|V(G)|} such that for two distinct vertices x and y, ω(x)≠ω(y), where ω(x)=∑y∈N(x)f(y), for N(x) the open neighborhood of x. It was conjectured that a graph G is distance antimagic if and only if G contains no two vertices with the same open neighborhood. In this paper, we study several distance antimagic product graphs. The products under consideration are the three fundamental graph products (Cartesian, strong, direct), the lexicographic product, and the corona product. We investigate the consequence of the non-commutative (or sometimes called non-symmetric) property of the last two products to the antimagicness of the product graphs.

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“…Any graph that admits such a labeling is called an (a, d)distance antimagic graph. Further the condition is relaxed in [16] and defined that if w G (u) = w G (v), for any two distinct vertices of G, then f is called as distance antimagic labeling of G. The following problem was posted in [16].…”

Section: Introductionmentioning

confidence: 99%