Antimagic Labeling of Regular Graphs

Chang, Fei-Huang; Liang, Yu-Chang; Pan, Zhishi; Zhu, Xuding

doi:10.1002/jgt.21905

Cited by 77 publications

(50 citation statements)

References 10 publications

(12 reference statements)

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“…For regular graphs, Cranston, Liang, and Zhu [3] proved that every odd regular graph is antimagic. Later, Chang, Liang, Pan, and Zhu [2] proved the even case.…”

Section: Introductionmentioning

confidence: 94%

“…. , C * t with a given orientation D Result: A bijection c : A(D) → [dn] 1 for i = 1 to s do 2 Label v i n i v i 1 with i and assign the two numbers s + 2i − 1, s + 2i to the 3 two arcs in increasing order along the orientation of P i 1 ; 4 end 5 Denote by L the set of edges which have been labelled as above ;…”

Section: Now We Have the Desired Orientation Of All Cmentioning

confidence: 99%

See 1 more Smart Citation

A note on antimagic orientations of even regular graphs

Yang

2019

Discrete Applied Mathematics

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Motivated by the conjecture of Hartsfield and Ringel on antimagic labelings of undirected graphs, Hefetz, Mütze, and Schwartz initiated the study of antimagic labelings of digraphs in 2010. Very recently, it has been conjectured in [Antimagic orientation of even regular graphs, J. Graph Theory, 90 (2019), 46-53.] that every graph admits an antimagtic orientation, which is a strengthening of an earlier conjecture of Hefetz, Mütze and Schwartz. In this paper, we prove that every 2d-regular graph (not necessarily connected) admits an antimagic orientation, where d ≥ 2. Together with known results, our main result implies that the above-mentioned conjecture is true for all regular graphs.

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“…For regular graphs, Cranston, Liang, and Zhu [3] proved that every odd regular graph is antimagic. Later, Chang, Liang, Pan, and Zhu [2] proved the even case.…”

Section: Introductionmentioning

confidence: 94%

Section: Now We Have the Desired Orientation Of All Cmentioning

confidence: 99%

A note on antimagic orientations of even regular graphs

Yang

2019

Discrete Applied Mathematics

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“…By Algorithm 2, observe that the edges are labeled in C 1 in the order of P 1 1,2 , P 1 1,3 , P 1 2,4 , P 1 3,5 , P 1 4,6 , ..., P 1 t 1 −2,t 1 , P 1 t 1 −1,t 1 by using the numbers in [1, n 1 ] with the increasingly order along the orientation of each path. If k ∈ [2,9], we modify the label order of some fixed paths based on the Algorithm 2 to define the bijections c 2 , ..., c 9 . That is, when k = 2, we label the edges in C 2 in the order of P 2 1,3 , P 2 1,2 , P 2 2,4 , P 2 3,5 , P 2 4,6 , ..., P 2 t 2 −2,t 2 , P 2 t 2 −1,t 2 by using the numbers in [n 1 + 1, n 2 ] with the increasingly order along the orientation of each path; when 3 ≤ k ≤ 8, for every i ∈ [3, k], we label the edges in C i in the order of P i 1,3 , P i 2,4 , P i 1,2 , P i 3,5 , P i 4,6 , ..., P i t i −2,t i , P i t i −1,t i by using the numbers in [n i−1 + 1, n i ] with the increasingly order along the orientation of each path; when k = 9, we label the edges in C 9 in the order of P 9 1,3 , P 9 2,4 , P 9 3,5 , P 9 1,2 , P 9 4,6 , ..., P 9 t 9 −2,t 9 , P 9 t 9 −1,t 9 by using the numbers in [n 8 + 1, n 9 ] with the increasing order along the orientation of each path.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…Cranston [3] proved that regular bipartite graphs are antimagic. Chang et al [2] discussed the antimagic labeling of regular graphs. Cranston et al [4] proved that regular graphs of odd degree are antimagic.…”

Section: Introductionmentioning

confidence: 99%

Antimagic orientations of disconnected even regular graphs

Chen

Hao

2019

Discrete Mathematics

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A labeling of a digraph D with m arcs is a bijection from the set of arcs of D to {1, 2, . . . , m}. A labeling of D is antimagic if no two vertices in D have the same vertexsum, where the vertex-sum of a vertex u ∈ V (D) for a labeling is the sum of labels of all arcs entering u minus the sum of labels of all arcs leaving u. An antimagic orientation D of a graph G is antimagic if D has an antimagic labeling. Hefetz, Mütze and Schwartz in [J. Graph Theory 64(2010)219-232] raised the question: Does every graph admits an antimagic orientation? It had been proved that for any integer d, every 2d-regular graph with at most two odd components has an antimagic orientation. In this paper, we consider the 2d-regular graph with many odd components. We show that every 2d-regular graph with any odd components has an antimagic orientation provide each odd component with enough order.

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“…Cranston [4] proved that any -regular bipartite graph with ≥ 2 is antimagic. For nonbipartite regular graphs, Cranston et al [5] proved that every odd regular graph is antimagic, and later Bérczi et al [2], and Chang et al [3], independently, proved that every even regular graph is antimagic. For more information on antimagic labelings of graphs and related labeling problems, see the recent informative survey [7].…”

Section: Introductionmentioning

confidence: 99%

Antimagic orientations of even regular graphs

Song

Wang

et al. 2018

Journal of Graph Theory

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A labeling of a digraph D with m arcs is a bijection from the set of arcs of D to {1,…,m}. A labeling of D is antimagic if no two vertices in D have the same vertex‐sum, where the vertex‐sum of a vertex u∈V(D) for a labeling is the sum of labels of all arcs entering u minus the sum of labels of all arcs leaving u. Motivated by the conjecture of Hartsfield and Ringel from 1990 on antimagic labelings of graphs, Hefetz, Mütze, and Schwartz [On antimagic directed graphs, J. Graph Theory 64 (2010) 219–232] initiated the study of antimagic labelings of digraphs, and conjectured that every connected graph admits an antimagic orientation, where an orientation D of a graph G is antimagic if D has an antimagic labeling. It remained unknown whether every disjoint union of cycles admits an antimagic orientation. In this article, we first answer this question in the positive by proving that every 2‐regular graph has an antimagic orientation. We then show that for any integer d≥2, every connected, 2d‐regular graph has an antimagic orientation. Our technique is new.

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Antimagic Labeling of Regular Graphs

Cited by 77 publications

References 10 publications

A note on antimagic orientations of even regular graphs

A note on antimagic orientations of even regular graphs

Antimagic orientations of disconnected even regular graphs

Antimagic orientations of even regular graphs

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