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 $φ: E \rightarrow L$. We say that $φ$ is an antimagic $L$-labeling if the induced vertex sum $φ_+: V \rightarrow \mathbb {R}$ defined as $φ_+(u)=\sum_{uv\in E}φ(uv)$ is injective. Similarly, $φ$ is a product antimagic $L$-labeling of $G$ if the induced vertex product $φ_{\circ}: V \rightarrow \mathbb {R}$ defined as $φ_{\circ}(u)=\prod_{uv\in E}φ(uv)$ 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 $m$, $m \geq 3$, 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.