On the sigma chromatic number of the zero-divisor graphs of the ring of integers modulo n

Abstract

The zero-divisor graph of a commutative ring R with unity is the graph Γ(R) whose vertex set is the set of nonzero zero divisors of R, where two vertices are adjacent if and only if their product in R is zero. A vertex coloring c : V (G) → ℕ of a non-trivial connected graph G is called a sigma coloring if σ(u) = σ(ν) for any pair of adjacent vertices u and v. Here, σ(χ) denotes the sum of the colors assigned to vertices adjacent to x. The sigma chromatic number of G, denoted by σ(G), is defined as the least number of colors needed to construct a sigma coloring of G. In this paper, we analyze the structure of the zero-divisor graph of rings ℤn, where n = pn1 1 P2 n2 …Pm nm, where m,ni,n2, …,nm are positive integers and p1,p2, …,pm are distinct primes. The analysis is carried out by partitioning the vertex set of such zero-divisor graphs and analyzing the adjacencies, cardinality, and the degree of the vertices in each set of the partition. Using these properties, we determine the sigma chromatic number of these zero-divisor graphs.