  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 $m$, $m
\geq 3$, 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.
