A SUPERMAGIC LABELING OF FINITE COPIES OF CARTESIAN PRODUCT OF CYCLES

Abstract

A homomorphism of a graph H onto a graphG is defined to be a surjective mapping ψ : V (H) → V (G) such that whenever u,v are adjacent in H, ψ(u),ψ(v) are adjacent inG, that is the induced mapping ¯ ψ : E(H) → E(G) satisfying: if e is an edge of H with end vertices u and v, then ¯ ψ(e) is an edge ofG with end vertices ψ(u) andψ(v). A homomorphism ψ is harmonious if ¯ ψ is a bijection. A triplet [H,ψ,t] is called a supermagic frame of G if ψ is a harmonious homomorphism of H onto G and t : E(H) →{ 1,2,...,|E(H)|} is an injective mapping suchthat u∈ψ−1(v) t∗(u) is independent of the vertex v ∈ V (G). Note thatt ∗(u) is the sum oft(uw) wherew is adjacent to u. In 2000, Ivanˇco proved that if there is a supermagic frame of a graph G, thenG is supermagic. In this paper, we construct a supermagic frame of m(≥ 2) copies of Cartesian product of cycles and apply the Ivanˇco’s result to show that m copies of Cartesian product of cycles is supermagic.