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

TL;DR

This paper investigates special labelings called $H$-supermagic labelings in graphs, proving their existence for certain classes like firecrackers, banana trees, and flowers, under specific conditions.

Contribution

It establishes the existence of $H$-supermagic labelings for various graph classes, extending the theory of graph labelings with new constructions and results.

Findings

Firecracker graphs $F_{k,n}$ are $F_{2,n}$-supermagic for odd $n$

Banana trees $B_{k,n}$ are $B_{1,n}$-supermagic

Flowers $F_n$ are $C_3$-supermagic

Abstract

A simple graph $G=(V,E)$ admits an $H$-covering if every edge in $E$ is contained in a subgraph $H'=(V',E')$ of $G$ which is isomorphic to $H$. In this case we say that $G$ is $H$-supermagic if there is a bijection $f:V\cup E\to\{1,\ldots\lvert V\rvert+\lvert E\rvert\}$ such that $f(V)=\{1,\ldots,\lvert V\rvert\}$ and $\sum_{v\in V(H')}f(v)+\sum_{e\in E(H')}f(e)$ is constant over all subgraphs $H'$ 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.