Antimagic orientations of graphs with given independence number

Abstract

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.

Antimagic orientation of graphs with minimum degree at least 33

“…Particulary, they showed that every orientation of stars (other than K 1,2 ), wheels, and complete graphs (other than K 3 ) is antimagic. Conjecture 2 has been also verified for regular graphs [9,12,14,16], biregular bipartite graphs with minimum degree at least two [13], Halin graphs [19], graphs with large maximum degree [17], and graphs with large independence number [15]. In this paper, by supporting Conjecture 2, we obtain the results below.…”

“…The following result was proved in [15] without the furthermore part. However, the furthermore part is easy to obtain by following the same proof of Lemma 2.2 in [15] by just letting vertices in T to be not the endvertices of the edge-disjoint trails that decompose E(G), which can be definitely guaranteed by the conditions imposed on T . So we omit the proof.…”

“…This case basically follows the same idea as in the Proof of Theorem 1.5 in [15], but we repeat the process for self-completeness.…”

The antimagic orientation problems for graphs obtained by some graph operations

“…Using the Euler circuit, one can estimate the vertex sum of each vertex induced by the circuit. Our first lemma below is almost the same as Lemma 2.1 in [9], Lemma 2.2 in [19], and Lemma 7 in [14]. For the sake of completeness, we present the proof in the note.…”

“…There are several families of graphs proved to satisfy Conjecture 1.2 by different groups of researchers: K n with n ≥ 4, S n with n ≥ 3, W n with n ≥ 3, the (2d + 1)-regular graphs with d ≥ 0 by Hefetz et al [5]; the 2d-regular graphs with d ≥ 2 by Li, Song, Wang, Yang, and Zhang [9], by Yang [22], and by Song and Hao [18]; the biregular graphs by Shan and Yu [15]; the Halin graphs by Yu, Chang, and Zhou [24]; the caterpillars by Lozano [11]; the lobsters by Gao and Shan [7]; and the complete k-ary trees by Song and Hao [17]. In addition to the above special graphs, Yang, Carson, Owens, Perry, Singgih, Song, Zhang, and Xhang [23], proved that every connected graph with at least n ≥ 9 vertices and maximum degree at least n − 5 admits an antimagic labeling, and Song, Yang, and Zhang [19] proved that every graph G with independent number at most 4 or least |V (G)|/2 admits an antimagic orientation. Some of the above results are also true for disconnected graphs.…”