TOTAL PERFECT CODES IN GRAPHS REALIZED BY COMMUTATIVE RINGS

Let R be a commutative ring with unity not equal to zero and let Γ(R) be a zero-divisor graph realized by R. For a simple, undirected, connected graph G = (V, E), a total perfect code denoted by C(G) in G is a subset C(G) ⊆ V (G) such that |N(v) ∩ C(G)| = 1 for all v ∈ V (G), where N(v) denotes the open neighbourhood of a vertex v in G. In this paper, we study total perfect codes in graphs which are realized as zero-divisor graphs. We show a zero-divisor graph realized by a local commutative ring with unity admits a total perfect code if and only if the graph has degree one vertices. We also show that if Γ(R) is a regular graph on |Z ^∗ (R)| number of vertices, then R is a reduced ring and |Z ^∗ (R)| ≡ 0(mod 2), where Z ^∗ (R) is a set of non-zero zero-divisors of R. We provide a characterization for all commutative rings with unity of which the realized zero-divisor graphs admit total perfect codes. Finally, we determine the cardinality of a total perfect code in Γ(R) and discuss the significance of the study of total perfect codes in graphs realized by commutative rings with unity.