A novel approach to find partitions of zm with equal sum subsets via complete graphs

Abstract

In mathematics and computer sciences, the partitioning of a set into two or more disjoint subsets of equal sums is a well-known NP-complete problem, also referred to as partition problem. There are various approaches to overcome this problem for some particular choice of integers. Here, we use quadratic residue graph to determine the possible partitions of positive integers m = 2β, qβ, 2βq, 2qβ, qp, where p, q are odd primes and β is any positive integer. The quadratic residue graph is defined on the set Zm = {0, 1, · · ·, m − 1}, where Zm is the ring of residue classes of m, i.e., there is an edge between x, y ∈ Zm if and only if x2 ≡ y2 (mod m). We characterize these graphs in terms of complete graph for some particular classes of m.