Strong Parity Weighted Totally Antimagic Total (SPAT) Graphs

An 1−1 correspondence mapping λ : V (G) ∪E(G) → {1, 2, 3, . . . , p+q} is a total labeling of a finite undirected graph G without loops and multiple edges, where p = |V (G)| and q = |E(G)|. A Perfectly Antimagic Total (PAT) labeling is a Totally Antimagic Total (TAT) labeling in which each vertex weight is also pairwise distinct from each of its edge weights. In this paper, we introduce a new parameter called strong parity weighted labeling. A TAT labeling is a strong parity weighted TAT (SPAT) labeling if all the vertex (edge) weights are distinct even (odd) integers. Agraph that admits such labeling is called a strong parityweighted TAT (SPAT) graph. Our findings established that several well-known families of graphs, including cycles, paths, stars, complete graphs, bi-stars, and ladders, admit SPAT labeling. We first illustrate the SPAT labeling for these families utilizing existing methodologies in labeling theory. Furthermore, we develop novel techniques that extend the analysis to other graph families, determining their potential to admit SPAT labeling.