Antimagic labeling for subdivisions of graphs

An antimagic labeling for a graph G with m edges is a bijection between the edge set of G and {1, 2, . . . , m} such that when summing up the labels of all edges incident to the same vertex, different vertices will have different sums. A graph admits such a labeling is said to be antimagic. It was conjectured by Hartsfield and Ringel that every connected graph other than an edge is antimagic. In this paper, we study the subdivisions of graphs. By G(s), we mean the graph obtained by replacing each edge of G with a path on sedges. For various types of graphs, we give the conditions on the minimum degree of G and the numbers to show that G(s) is antimagic. Particularly, when G is a complete graph or a complete bipartite graph, we show that G(s) is antimagic for all s >= 2. Some different variations of the antimagic problem are studied and the corresponding results are presented in this paper as well. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.