Abstract
Let G be a graph with m edges. A product antimagic labeling of G is a bijection from the edge set E G to the set 1,2 , … , m such that the vertex-products are pairwise distinct, where the vertex-product of a vertex v is the product of labels on the incident edges of v . A graph is called product antimagic if it admits a product antimagic labeling. In this paper, we will show that caterpillars with at least three edges are product antimagic by an O m log m algorithm.
Highlights
Let G be a graph with m edges. e vertex set and edge set of G are denoted by V(G) and E(G), respectively
In [8], the authors proved that paths with at least four vertices and 2-regular graphs are product antimagic
Kaplan et al [9] proved that the following graphs are product antimagic: the disjoint union of cycles and paths where each path has at least three edges; connected graphs with n vertices and m edges where m ≥ 4nlnn; graphs G where each component has at least t w o e d g e s and the minimum degree of G is at least 8 ln|E|ln(ln|E|); and all complete k-partite graphs except K2 and K1,2
Summary
They conjectured that every connected graph other than K2 is antimagic. In [8], the authors proved that paths with at least four vertices and 2-regular graphs are product antimagic. A connected graph with at least three edges is product antimagic. Kaplan et al [9] proved that the following graphs are product antimagic: the disjoint union of cycles and paths where each path has at least three edges; connected graphs with n vertices and m edges where m ≥ 4nlnn; graphs G where each component has at least t w o e d g e s and the minimum degree of G is at least 8 ln|E|ln(ln|E|); and all complete k-partite graphs except K2 and K1,2. We prove that Conjecture 1 is affirmative for caterpillars with at least three edges.
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