On Path-Supermagic Labelings of Cycles

Abstract

A graph $G$ admits an $H$-covering if every edge in $E(G)$ belongs to a subgraph of $G$ isomorphic to $H$. The graph $G$ is said to be $H$-magic if there exists a bijection $f$ from $V(G)\cup E(G)$ to $\{1,2,\dots ,|V(G)|+|E(G)|\}$ such that for every subgraph $H^{\prime }$ of $G$ isomorphic to $H$, $\sum _{v\in V(H^{\prime })}f(v)+\sum _{e\in E(H^{\prime })}f(e)$ is constant. When $f(V(G))=\{1,2,\dots ,|V(G)|\}$, then $G$ is said to be $H$-supermagic. In this paper, we investigate path-supermagic cycles. We prove that for two positive integers $m$ and $t$ with $m>t\ge 2$, if $C_m$ is $P_t$-supermagic, then $C_{3m}$ is also $P_t$-supermagic. Moreover, we show that for $t\in \{3,4,9\}$, $C_n$ is $P_t$-supermagic if and only if $n$ is odd with $n>t$.