Antimagic orientations of disconnected even regular graphs

Abstract

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.

Antimagic orientation of graphs with minimum degree at least 33

“…Particularly, they showed that every orientation of stars (other than ), wheels, and complete graphs (other than ) 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.…”

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.…”

Antimagic orientation of subdivided caterpillars

“…In [7], Hefetz, Mütze and Schwartz proved Conjecture 1 for some classes of graphs, such as stars, wheels, and graphs of order n with minimum degree at least c log n for an absolute constant c. In the process the authors proved a stronger result that every orientation of these graphs is antimagic as well. Additional cases for this conjecture that have been proved already include regular graphs [7,9,16,15], biregular bipartite graphs with minimum degree at least two [13], Halin graphs [18], graphs with large maximum degree [17], graphs with minimum degree at least 33 and bipartite graphs with no vertex of degree 0 or 2 [12]. Researchers have taken particular interest in investigating trees, as we do.…”