On Super \((a, 1)\)-Edge-Antimagic Total Labelings of Grids and Crowns

Abstract

A graph \(G(V,E)\) with order \(p\) and size \(q\) is called \((a,d)\)-edge-antimagic total labeling graph if there exists a bijective function \(f : V(G) \cup E(G) \rightarrow \{1, 2, \ldots, p+q\}\) such that the edge-weights \(\lambda_{f}(uv) = f(u) + f(v) + f(uv)\), \(uv \in E(G)\), form an arithmetic sequence with first term \(a\) and common difference \(d\). Such a labeling is called super if the \(p\) smallest possible labels appear at the vertices. In this paper, we study super \((a, 1)\)-edge-antimagic properties of \(m(P_{4} \square P_{n})\) for \(m, n \geq 1\) and \(m(C_{n} \odot \overline{K_{l}})\) for \(n\) even and \(m, l \geq 1\).