Super \((a, 1)\)-cycle-antimagic labeling of the grid

Abstract

Let \(G\) and \(F\) be graphs. If every edge of \(G\) belongs to a subgraph of \(G\) isomorphic to \(F\), and there exists a bijection \(\lambda: V(G) \bigcup E(G) \rightarrow \{1, 2, \ldots, |V(G)| + |E(G)|\}\) such that the set \(\{\sum_{v\in V(F’)}\lambda(v)+\sum_{e\in E(f’)}\lambda(e):F’\cong F,F’\subseteq G\}\) forms an arithmetic progression starting from \(a\) and having common difference \(d\), then we say that \(G\) is \((a,d)\)-\(F\)-antimagic. If, in addition, \(\lambda(V(G)) = \{1, 2, \ldots, |V(G)|\}\), then \(G\) is \emph{super} \((a,d)\)-\(F\)-antimagic. In this paper, we prove that the grid (i.e., the Cartesian product of two nontrivial paths) is super \((a,1)\)-\(C_4\)-antimagic.