Antimagic and product antimagic graphs with pendant edges

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 → L. We say that φ is an antimagic L-labeling if the induced vertex sum φ₊: V → R defined as φ₊(u)=∑uv∈ Eφ(uv) is injective. Similarly, φ is a product antimagic L-labeling of G if the induced vertex product φ∘: V → R defined as φ∘(u)=∏uv∈ Eφ(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,,m\. Hartsfield and Ringel conjectured that every simple connected graph distinct from K₂ is antimagic, but the conjecture remains widely open. We prove, among other results, that every connected graph of size m, m ≥ 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.