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 $\lambda :V(G)\bigcup E(G)\to \{1,2,\dots ,|V(G)|+|E(G)|\}$ such that the set $\{\sum _{v\in V(F^{\prime })}\lambda (v)+\sum _{e\in E(f^{\prime })}\lambda (e):F^{\prime }\cong F,F^{\prime }\subseteq G\}$ 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, $\lambda (V(G))=\{1,2,\dots ,|V(G)|\}$, 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.