$H$-supermagic labelings for firecrackers, banana trees and flowers

Abstract

A simple graph $G=(V,E)$ admits an $H$-covering if every edge in $E$ is contained in a subgraph $H^{\prime }=(V^{\prime },E^{\prime })$ of $G$ which is isomorphic to $H$. In this case we say that $G$ is $H$-supermagic if there is a bijection $f:V\cup E\to \{1,\dots |V|+|E|\}$ such that $f(V)=\{1,\dots ,|V|\}$ and ${\sum }_{v\in V(H^{\prime })}f(v)+{\sum }_{e\in E(H^{\prime })}f(e)$ is constant over all subgraphs $H^{\prime }$ of $G$ which are isomorphic to $H$. In this paper, we show that for odd $n$ and arbitrary $k$, the firecracker $F_{k,n}$ is $F_{2,n}$-supermagic, the banana tree $B_{k,n}$ is $B_{1,n}$-supermagic and the flower $F_n$ is $C_3$-supermagic.