Antimagic graph constructions with triangle and three-path unions

Abstract

Let $G=(V,E)$ be a graph with $p$ edges and let $f$ be a bijective function from $E(G)$ to $\{1,2,\dots ,p\}$.
For any vertex $v$, let $\phi_f(v)$ denote the sum of $f(e)$ over all edges $e$ incident to $v$.
If $\phi_f(v)\not=\phi_f(u)$ holds for any two distinct vertices $u$ and $v$, then $f$ is called an antimagic labelling of $G$.
A graph $G$ is deemed antimagic if it admits such a labelling.
In this study, we investigate the antimagic properties of graph unions, particularly focusing on structures composed of multiple triangles and 3-paths. We employ Skolem sequences and extended Skolem sequences to construct antimagic labelling for these graph unions. Specifically, we demonstrate that for any integer $n\geq 9$, the graph formed by the disjoint union of $m$ copies of the triangle $C_3$ and $n$ copies of the path $P_3$ is antimagic for $m\geq \lceil\frac{n}{3}\rceil$.