On Path-Supermagic Labelings of Cycles

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 \( H’ \) of \( G \) isomorphic to \( H \), \( \sum_{v\in V(H’)} f(v) + \sum_{e\in E(H’)} f(e) \) 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 \( m > t \geq 2 \), 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 \( n > t \).