Constructions of $H$-Antimagic Graphs Using Smaller Edge-Antimagic Graphs

Abstract

A simple graph $G=(V,E)$ admits an $H$-covering if every edge in $E$ belongs to at least one subgraph of $G$ isomorphic to a given graph $H$. An $(a,d)$-$H$-antimagic labeling of $G$ admitting an $H$-covering is a bijective function $f:V\cup E\to \{1,2,\dots ,|V|+|E|\}$ such that, for all subgraphs $H^{\prime }$ of $G$ isomorphic to $H$, the $H^{\prime }$-weights, $wt(H^{\prime })=\sum _{v\in V(H^{\prime })}f(v)+\sum _{e\in E(H^{\prime })}f(e)$, constitute an arithmetic progression with the initial term $a$ and the common difference $d$. Such a labeling is called super if $f(V)=\{1,2,\dots ,|V|\}$. In this paper, we study the existence of super $(a,d)$-$H$-antimagic labelings for graph operation $G^H$, where $G$ is a (super) $(b,d^{\ast })$-edge-antimagic total graph and $H$ is a connected graph of order at least $3$.