This research studies the chromatic numbers of the suborbital graphs for the modular group and the extended modular group. We verify that the chromatic numbers of the graphs are2or3. The forest conditions of the graphs for the extended modular group are also described in this paper.

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This research introduces a gcd-pair in Z n which is an unordered pair {[a] n , [b] n } of elements in Z n such that 0 ≤ a, b < n and the greatest common divisor gcd(a, b) divides n. The properties of gcd-pairs in Z n and their graph representations are investigated. We also provide the counting formula of gcd-pairs in Z n and its subsets. The algorithms to find, count and check gcd-pairs in Z n are included. MSC: 11A05,