Antimagic Labeling of Graph Unions of Trees and 4-Cycles

Abstract

Let $G(V,E)$ be a graph with $V(G)$ as the set of vertices and $E(G)$ as the set of edges. A labeling is a bijection $f:E(G)\to\{1,2,\cdots, |E(G)|\}$. For each vertex \(v\), let \(\phi(v)\) be the sum of the labels on the edges incident to \(v\). If all values \(\phi(v)\) are distinct, the labeling is antimagic, and the graph is antimagic if such a labeling exists. This paper explores the concept of antimagic labeling in graph theory, with a particular focus on the union of various tree structures, including stars, single brooms, and double brooms. We apply extended Skolem sequences to prove that for a wide range of parameters, unions of multiple 3-paths with appropriately many 4-cycles yield antimagic graphs. Additionally, we analyze combinations of 4-cycles paired with different tree structures.