Structural and Hamiltonian Properties of the <b><i>n</b></i>–Array Unit Graphs Over Finite Rings

Consider a finite ring R, and let U( R ) denote the set of all its unit elements. The unit graph G_R of R is defined as follows: the vertex set V( G_R ) consists of all elements of R. Two distinct elements a,\ b∈ \ G_R are adjacent if and only if a + b∈ U( R ). Recent research has provided necessary and sufficient conditions for the unit graph G_R to have a Hamiltonian cycle. However, the property of Hamiltonicity under direct product constructions of rings has not been fully understood. This paper addresses this gap by studying whether the unit graph G_R^n of the ring $R^n = \ \{ ( r1,\ r2, \ldots , r_n )\mid \ r_i\in \ R,1 \le \ i \le \ n \}$ admits a Hamiltonian cycle, assuming G_R does. To this end, the n - array unit graph of R, denoted G_R^n, is defined and proved to be isomorphic to G_R^n. A counterexample is presented to demonstrate that the existence of a Hamiltonian cycle in G_R does not, in general, imply Hamiltonicity in G_R^n. Furthermore, necessary and sufficient conditions are derived for G_R^n to have a Hamiltonian cycle, the set $U^n( R ) = \{ ( r1,r2,\ldots ,r_n )^T\mid r_i\in U( R )\} $ forms a dominating set in G_R^n. In addition to Hamiltonicity, the coloring properties of G_R^n are also examined. Upper bounds for the chromatic number χ ( G_R^n) ) are derived, including the general bound χ ( G_R^n ) ≤ | U( R ) |^n\ + \ 1. Moreover, under the assumption that R is generated by its unit elements, this bound is improved to χ ( G_R^n ) ≤ | U( R ) |^n. These results clarify the limitations of extending Hamiltonian properties to unit graphs of direct product rings and contribute to a deeper structural understanding of Hamiltonicity and coloring in higher-dimensional unit graphs.