Every graph is homeomorphic to an antimagic bipartite graph (Q6564831)

A labeling of a graph $G$ is a function from $E(G)$ to a set of non-negative integers. A labeling of $G$ is antimagic if it is a bijection from $E(G)$ to $\{1, 2, \dots,|E(G)|\}$ such that all vertex sums are pairwise distinct, where the vertex sum at vertex $v$ (denoted by $s_G (v)$) is the sum of the labels assigned to the edges incident to $v$. A graph is called antimagic if the graph resulting from deleting its isolated vertices admits an antimagic labeling.\N\NThe theoretical framework of this paper is based on the concepts of antimagic labeling of graphs and homeomorphic graphs, as well as the Hartsfield-Ringel conjecture on antimagic graphs. The main finding of this paper is ``Every graph is homeomorphic to an antimagic bipartite graph, and consequently, every graph is homeomorphic to a graph that admits an antimagic orientation''. They have also shown that every tree is homeomorphic to infinitely many antimagic trees. This result highlights the richness of tree structures in the context of antimagic labeling. The authors opted for the methodology which consists of a 3-step algorithm to construct a bipartite graph $G^{\ast}$ that is homeomorphic to the original graph $G$.