Characterization of product anti-magic graphs of large order (Q2478164)

A graph \(G=(V,E)\) is called to be product anti-magic if there is a labeling of edges of \(G\) by labels \(1,\dots,\left| E\right| \) so that no two vertices have the same product of incident labels. It has been conjectured that each connected graph of order \(n\geq 4\) is product anti-magic. The author verifies this conjecture for all connected graphs of order \(n_{0}\geq 10^{10^{20}}\). In fact he shows that all graphs of order \(n\geq n_{0},\) except for graphs (i) with an isolated edge, (ii) with at least two isolated vertices, (iii) unions of vertex disjoint \( K_{1,2},\) and (iv) consisting of one isolated vertex and vertex disjoint spiders (a star with some of its edges subdivided by a vertex) are product anti-magic.