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: E L. We say that is an antimagic L-labeling if the induced vertex sum _+: V R defined as _+ (u) =ₔₕ ₄ (uv) is injective. Similarly, is a product antimagic L-labeling of G if the induced vertex product _: V R defined as _ (u) =ₔₕ ₄ (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.