Antimagic orientation of graphs with minimum degree at least 33

Abstract

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 {1, • • • , m} 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 G has an antimagic orientation if it has an orientation which 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 without both isolated and degree 2 vertices admits an antimagic orientation and every graph G with δ(G) ≥ 33 admits an antimagic orientation. Our proof relies on a newly developed structural property of bipartite graphs, which might be of independent interest.

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.…”

“…We again call the corresponding subdivision of the spine of T the spine of T * , and call the corresponding subdivisions of the legs of T the legs of T * . It was proved in [12] that every bipartite graph without vertices of degree 0 or 2 admits an antimagic orientation. It suggests that constructing antimagic orientations for graphs with many vertices of degree 2 is very difficult in general.…”

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.…”