Totally antimagic total labeling of helm and gear graphs - Earl Baron Marzan Almanzor; Michael Kirby Briones Rodriguez (Indonesian Journal of Combinatorics, 2026)
A total labeling of a graph G is a bijection from the union of the vertex set and the edge set of G to the set {1,2,...,|V(G)|+|E(G)|}. Under a total labeling, the vertex-weight of a vertex is defined as the sum of its label and the labels of all edges incident to it. Similarly, the edge-weight of an edge is the sum of its label and the labels of its two end vertices. A total labeling is said to be edge-antimagic total if all the edge-weights are pairwise distinct, and vertex-antimagic total if all the vertex-weights are pairwise distinct. If a total labeling is edge-antimagic total and vertex-antimagic total at the same time, then it is called a totally antimagic total labeling. A graph that admits totally antimagic total labeling is called a totally antimagic total graph. In this paper, we show that helm graphs Hn and gear graphs Gn are totally antimagic total graphs.