Let \(G=(V,E)\) be a simple graph of size m and L a set of m distinct real numbers. An L-labeling of G is a bijection \(\phi : E \rightarrow L\) . We say that \(\phi \) is an antimagic L-labeling if the induced vertex sum \(\phi _+: V \rightarrow {\mathbb {R}}\) defined as \(\phi _+(u)=\sum _{uv\in E}\phi (uv)\) is injective. Similarly, \(\phi \) is a product antimagic L-labeling of G if the induced vertex product \(\phi _{\circ }: V \rightarrow {\mathbb {R}}\) defined as \(\phi _{\circ }(u)=\prod _{uv\in E}\phi (uv)\) is injective. A graph G is antimagic (resp. product antimagic) if it has an antimagic (resp. a product antimagic) L-labeling for \(L=\{1,2,\dots ,m\}\) . Hartsfield and Ringel conjectured that every simple connected graph distinct from \(K_2\) is antimagic, but the conjecture remains widely open. We prove, among other results, that every connected graph of size m, \(m \ge 3\) , admits an antimagic L-labeling for every arithmetic sequence L of m positive real numbers, if every vertex of degree at least three is a support vertex. As a corollary, we derive that these graphs are antimagic, reinforcing the veracity of the conjecture by Hartsfield and Ringel. Moreover, these graphs admit also a product antimagic L-labeling provided that the smallest element of L is at least one. The proof is constructive.
