A SUPERMAGIC LABELING OF FINITE COPIES OF CARTESIAN PRODUCT OF CYCLES

Abstract

A homomorphism of a graph H onto a graph G 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 in G, 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 of G with end vertices ψ(u) and ψ(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 such

that

u∈ψ−1(v)

t ∗(u) is independent of the vertex v ∈ V (G). Note that

t ∗(u) is the sum of t(uw) where w is adjacent to u.

In 2000, Ivanˇco proved that if there is a supermagic frame of a graph

G, then G 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.