Graceful labeling of zero-divisor graph Γ(ℤp²q) and Γ(ℤp³q)

Abstract

Some papers have already provided graceful labeling for some types of zero-divisor graphs. We reviewed the graceful labeling results of Γ(ℤ₂₅), Γ(ℤ₈), and Γ(ℤ₂₇), then use those results to label zero-divisors graphs Γ(ℤ_25q), Γ(ℤ_8q), and Γ(ℤ_27q).

The result is that there is graceful labeling for Γ(ℤ_p²q) for p = 5 and Γ(ℤ_p³q) for p = 2, 3, where q is prime number that is different from p. In this paper, we provide the graceful labeling of zero-divisor graph Γ(ℤ_p²q) and Γ(ℤ_p³q) with adaptation and modification of existing results.