An Inductive Approach to Strongly Antimagic Labelings of Graphs

Abstract

An antimagic labeling for a graph $G$ with $m$ edges is a bijection $f: E(G) \to \{1, 2, \dots, m\}$ so that $\phi_f(u) \neq \phi_f(v)$ holds for any pair of distinct vertices $u, v \in V(G)$, where $\phi_f(x) = \sum_{x \in e} f(e)$. A strongly antimagic labeling is an antimagic labeling with an additional condition: For any $u, v \in V(G)$, if $\deg(u) > \deg(v)$, then $\phi_f(u) > \phi_f(v)$. A graph $G$ is strongly antimagic if it admits a strongly antimagic labeling. We present inductive properties of strongly antimagic labelings of graphs. This approach leads to simplified proofs that spiders and double spiders are strongly antimagic, previously shown by Shang [Spiders are antimagic, Ars Combinatoria, 118 (2015), 367--372] and Huang [Antimagic labeling on spiders, Master's Thesis, Department of Mathematics, National Taiwan University, 2015], and by Chang, Chin, Li and Pan [The strongly antimagic labelings of double spiders, Indian J. Discrete Math. 6 (2020), 43--68], respectively. We fix a subtle error in [The strongly antimagic labelings of double spiders, Indian J. Discrete Math. 6 (2020), 43--68]. Further, we prove certain level-wise regular trees, cycle spiders and cycle double spiders are all strongly antimagic.