$P_h$-Supermagic Labelings of Some Trees

Abstract

Let $P_h$ be a path on $h$ vertices. A simple graph $G=(V,E)$ admits a $P_h$-covering if every edge in $E$ belongs to a subgraph of $G$ that is isomorphic to $P_h$. $G$ is called $P_h$-magic if there is a total labeling $f:V\cup E\to \{1,2,\dots ,|V|+|E|\}$ such that for each subgraph $H^{\prime }=(V^{\prime },E^{\prime })$ of $G$ that is isomorphic to $P_h$, $\sum _{v\in V^{\prime }}f(v)+\sum _{e\in E^{\prime }}f(e)$ is constant. When $f(V)=\{1,2,\dots ,|V|\}$, we say that $G$ is $P_h$-supermagic.

In this paper, we study some $P_h$-supermagic trees. We give some sufficient or necessary conditions for a tree to be $P_h$-supermagic. We also consider the $P_h$-supermagicness of special types of trees, namely shrubs and banana trees.