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

Garciano, Agnes D. (Ateneo de Manila University)
Marcelo, Reginaldo M. (Ateneo de Manila University)
Ruiz, Mari-Jo P. (Ateneo de Manila University)
Tolentino, Mark Anthony Cayanan (Ateneo de Manila University)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 13, No 2 (2025): Electronic Journal of Graph Theory and Applications

The objective of this paper is to investigate a particular graph coloring, called sigma coloring, as applied to ideal-based zero-divisor graphs. Given a commutative ring R with (nonzero) identity and a proper ideal I of R, the graph ΓI(R) is defined as an undirected graph with vertex set { x ∈ R \ I : xy ∈ I for some y ∈ R \ I } and edge set { xy : xy ∈ I }. On the other hand, given a graph G, a sigma coloring c: V(G) → ℕ is a coloring that satisfies σ(u) ≠ σ(v) for any two adjacent vertices u,v in G, where σ(x) denotes the sum of all colors c(y) among all neighbors y of a vertex x. The sigma chromatic number of G is denoted by σ(G) and is defined as the fewest number of colors needed for a sigma coloring of G. In this paper, we completely determine the sigma chromatic number of ideal-based zero-divisor graphs of rings of integers modulo n.