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Abstract

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.

Keywords

Graph from rings, Hamiltonian graph, Jacobson radical, n–Array Unit Graph, Unit graph ring

Subject Area

Mathematics

Article Type

Article

First Page

3249

Last Page

3261

Creative Commons License

Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.

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