A note on antimagic orientations of even regular graphs

Yang, Donglei

doi:10.1016/j.dam.2019.04.017

Cited by 16 publications

(10 citation statements)

References 9 publications

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“…Particularly, they showed that every orientation of stars (other than

K_{1, 2}

), wheels, and complete graphs (other than

K_{3}

) is antimagic. Conjecture 1 has been also verified for regular graphs [9,13,15,17], biregular bipartite graphs with minimum degree at least two [14], Halin graphs [20], graphs with large maximum degree [18], and graphs with large independence number [16]. In this paper, by supporting Conjecture 1 we obtain the results below.…”

Section: Introductionsupporting

confidence: 74%

Antimagic orientation of graphs with minimum degree at least 33

Shan

2021

Journal of Graph Theory

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An antimagic labeling of a directed graph D with n vertices and m arcs is a bijection from the set of arcs of D to the integers MathClass-open{ 1 , … , m MathClass-close} such that all n oriented vertex sums are pairwise distinct, where an oriented vertex sum is the sum of labels of all arcs entering that vertex minus the sum of labels of all arcs leaving it. A graph has an antimagic orientation if it has an orientation that admits an antimagic labeling. Hefetz, Mütze, and Schwartz conjectured that every connected graph admits an antimagic orientation. In this paper, we show that every bipartite graph with no vertex of degree 0 or 2 admits an antimagic orientation and every graph with minimum degree at least 33 admits an antimagic orientation.

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“…Particularly, they showed that every orientation of stars (other than

K_{1, 2}

), wheels, and complete graphs (other than

K_{3}

Section: Introductionsupporting

confidence: 74%

Antimagic orientation of graphs with minimum degree at least 33

Shan

2021

Journal of Graph Theory

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“…Conjecture 1.1 has been verified to be true for odd regular graphs [9], disjoint union of cycles or connected 2d-regular graphs with d ≥ 2 by Li, Song, Wang, Yang, and Zhang [11], and disconnected 2d-regular graphs with d ≥ 2 by Yang [13]. A bipartite graph G with bipartition (A, B) is (a, b)biregular if each vertex in A has degree a and each vertex in B has degree b. Shan and Yu [12] recently proved that every (a, b)-biregular bipartite graph admits an antimagic orientation.…”

Section: Conjecture 11 ([9]mentioning

confidence: 99%

Antimagic orientations of graphs with given independence number

Song

Yang

Zhang

2019

Preprint

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Given a digraph D with m arcs and a bijection τ : A(D) → {1, 2, . . . , m}, we say (D, τ ) is an antimagic orientation of a graph G if D is an orientation of G and no two vertices in D have the same vertex-sum under τ , where the vertex-sum of a vertex u in D under τ is the sum of labels of all arcs entering u minus the sum of labels of all arcs leaving u. Hefetz, Mütze, and Schwartz in 2010 initiated the study of antimagic orientations of graphs, and conjectured that every connected graph admits an antimagic orientation. This conjecture seems hard, and few related results are known. However, it has been verified to be true for regular graphs, biregular bipartite graphs, and graphs with large maximum degree. In this paper, we establish more evidence for the aforementioned conjecture by studying antimagic orientations of graphs G with independence number at least |V (G)|/2 or at most four. We obtain several results. The method we develop in this paper may shed some light on attacking the aforementioned conjecture.

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“…As a relaxation of this problem, Hefetz, Mütze, and Schwartz [9] proposed the following conjecture. Very recently, Conjecture 1.1 has been verified to be true for regular graphs (see [9,10,12]), and biregular bipartite graphs with minimum degree at least two by Shan and Yu [11]. Motivated by the work of Yilma [13], we establish more evidence for Conjecture 1.1 in this paper by studying antimagic orientations of graphs with large maximum degrees.…”

Section: Introductionmentioning

confidence: 62%

“…The main idea of the proofs of Theorem 1.2 and the preliminary results (stated and proved in Section 2) is to use Eulerian orientations. This strategy was first used in [9] and then in [10,12]. Our method here is more involved to obtain antimagic orientations of such graphs.…”

Section: Introductionmentioning

confidence: 99%

Antimagic orientations of graphs with large maximum degree

Yang

Carlson

Owens

et al. 2019

Preprint

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Given a digraph D with m arcs, a bijection τ : A(D) → {1, 2, . . . , m} is an antimagic labeling of D if no two vertices in D have the same vertex-sum, where the vertex-sum of a vertex u in D under τ is the sum of labels of all arcs entering u minus the sum of labels of all arcs leaving u. We say (D, τ ) is an antimagic orientation of a graph G if D is an orientation of G and τ is an antimagic labeling of D. Motivated by the conjecture of Hartsfield and Ringel from 1990 on antimagic labelings of graphs, Hefetz, Mütze, and Schwartz in 2010 initiated the study of antimagic orientations of graphs, and conjectured that every connected graph admits an antimagic orientation. This conjecture seems hard, and few related results are known. However, it has been verified to be true for regular graphs and biregular bipartite graphs. In this paper, we prove that every connected graph G on n ≥ 9 vertices with maximum degree at least n − 5 admits an antimagic orientation.

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A note on antimagic orientations of even regular graphs

Cited by 16 publications

References 9 publications

Antimagic orientation of graphs with minimum degree at least 33

Antimagic orientation of graphs with minimum degree at least 33

Antimagic orientations of graphs with given independence number

Antimagic orientations of graphs with large maximum degree

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