A Paradigmatic Approach to Find Equal Sum Partitions of Zero-Divisors via Complete Graphs

Abstract

In computer science and mathematics, a partition of a set into two or more disjoint subsets with equal sums is a well-known NP-complete problem. This is a hard problem and referred to as the partition problem or number partitioning. In this paper, we solve a particular type of NP-complete problem on the set of all zero-divisors of Zn including zero, where Zn is the ring of residue classes of a positive integer n. In this regard, we introduce and investigate quadratic zero-divisor graph in which we build an edge between zero-divisors zi and zj if and only if zi2≡zj2 mod n,i≠j. This is denoted as G⏞2,n. We characterize these graphs in term of complete graphs for classes of integers 2α,pα,2αp,2pα and pq, where α is any positive integer and p,q are odd primes.