Constructing the antimagic labelings for double stars union paths on three vertices

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.