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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 φ f (v) the sum of f (e) over all edges e incident to v. If φ f (v) = φ f (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 [8] conjectured that all connected graphs except P 2 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), let τ (G) be the maximum integer such that G ∪ tP 3 is antimagic for all t τ (G). The existence of finite τ (G) was presented by Chang, Chen, Li, and Pan [3]. Further, Shang, Lin, Liaw [16] and Li [12] showed tight bounds of τ (G) for star forests and balanced double stars, respectively. We generalize these results by proving an upper bound of τ (G) for all graphs without isolated vertices and P 2 as components. Besides star forests and balanced double stars, this general bound is also tight on many other families of graphs, including cycles C n , 3 n 9, double triangle 2C 3 and others. In addition, we discuss properties of τ (G) and pose open questions, including whether our bound is always tight.