Antimagic labeling for unions of graphs with many three-paths

Abstract

Let G be a graph with m edges and let f be a bijection from E(G) to {1,2,…,m}. For any vertex v, denote by ϕf(v) the sum of f(e) over all edges e incident to v. If ϕf(v)≠ϕf(u) holds for any two distinct vertices u and v, then f is called an antimagic labeling of G. We call G antimagic if such a labeling exists. Hartsfield and Ringel [10] conjectured that all connected graphs except P2 are antimagic. Denote the disjoint union of graphs G and H by G∪H, and the disjoint union of t copies of G by tG. For an antimagic graph G (connected or disconnected), we define the parameter τ(G) to be the maximum integer such that G∪tP3 is antimagic for all t⩽τ(G). Chang, Chen, Li, and Pan showed that for all antimagic graphs G, τ(G) is finite [3]. Further, Shang, Lin, Liaw [19] and Li [14] found the exact value of τ(G) for special families of graphs: star forests and balanced double stars, respectively. They did this by finding explicit antimagic labelings of G∪tP3 and proving a tight upper bound on τ(G) for these special families. In the present paper, we generalize their results by proving an upper bound on τ(G) for all graphs. For star forests and balanced double stars, this general bound is equivalent to the bounds given in [19] and [14] and tight. In addition, we prove that the general bound is also tight for every other graph we have studied, including an infinite family of jellyfish graphs, cycles Cn with 3⩽n⩽9, and the double triangle 2C3.