In this note, we study a family of subgraphs of the Farey graph, denoted as F N for every N ∈ N. We show that F N is connected if and only if N is either equal to one or a prime power. We introduce a class of continued fractions referred to as F N -continued fractions for each N > 1. We establish a relation between F N -continued fractions and certain paths from infinity in the graph F N . We discuss existence and uniqueness of F N -continued fraction expansions of real numbers.
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