Constructing the antimagic labelings for double stars union paths on three vertices
Wei-Tian Li, Po-Wen Yang
TL;DR
This paper analyzes antimagic labelings for the disjoint union of a double star $S_{a,b}$ with $c$ copies of $P_3$. It provides an exact determination of $\tau(S_{a,b})$, the maximum $c$ such that $S_{a,b}+cP_3$ is antimagic, and gives constructive labeling methods for all permitted $c$, including explicit schemes using sets of edge groups and a Pell-equation lens for $(1,1)$-antimagic labelings. It also demonstrates that the upper bounds can be strict in some cases, producing $\tau(S_{a,b})<\beta(S_{a,b})$, and reports several concrete $(1,1)$-antimagic examples (e.g., $S_{1,2}+5P_3$). The results advance the understanding of how disjoint unions with small paths interact with antimagic properties and connect to Diophantine equations via the Pell framework.
Abstract
For a graph on $m$ edges, a bijective function between the edge set of the graph and $\{1,2,\ldots,m\}$ is an antimagic labeling provided that when adding the labels of the edges incident to the same vertex, the sums are pairwise distinct. Hartsfield and Ringel conjectured that every connected graph has antimagic labeling. On the other hand, it is known that for any graph $G$, the disjoint union of $G$ and many $P_3$, a path on 3 vertices, is not antimagic. In this paper, we determined the exact number of $P_3$'s such that the disjoint union of a double star with the number of $P_3$'s is antimagic. In addition, we provide some examples of $(1,1)$-antimagic labelings. That is, the antimagic labelings have vertex sums 1 through the number of vertices of the graphs.