JP Journal of Algebra, Number Theory and Applications
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Abstract: A comaximal graph is
introduced in [7] and studied in [4]. In this paper, we investigate the
chromatic numbers of comaximal graphs and also of their complementary graphs.
Let Rbe a Noetherian
ring and let be the chromatic number of the
complementary graph to a comaximal graph If is finite, then Ris a field or a finite ring. Furthermore, the following assertions hold:
(1) If Ris a field, then
(2) If Ris a finite ring, then where are
all maximal ideals of Rand
denotes
the number of elements of the set for As for comaximal graphs, we give a
partial result on chromatic numbers.
Keywords and phrases: comaximal graph, zero-divisor, semi-local ring, chromatic number.
