JP Journal of Algebra, Number Theory and Applications
Volume 38, Issue 6, Pages 589 - 607
(December 2016) http://dx.doi.org/10.17654/NT038060589 |
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HAMILTONIAN PROPERTY OF A MAXIMAL GRAPH AND CHROMATIC NUMBER OF ITS LINE GRAPH
Arti Sharma and Atul Gaur
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Abstract: Let Rbe a commutative ring with identity. Let denote the maximal graph associated to R, that is, is a graph with vertices as non-units of R, where two distinct vertices aand bare adjacent if and only if there is a maximal ideal of Rcontaining both. In this paper, we study the cut vertices, Hamiltonian property of and chromatic number of line graph of denoted by For any ring R, it is shown that has a cut vertex if and only if where and are fields. The complete characterization for a graph Gto be a maximal graph for some ring R is given. We have also shown that, for any finite ring R, is Hamiltonian if either Rhas atleast three maximal ideals or where and are local rings but not a field. Finally, we have determined the chromatic number of
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Keywords and phrases: line graph, Hamiltonian graph, chromatic number. |
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