Abstract
A small-world Farey graph is a recursively constructed graph characterized as a small-world network graph. Many of the network properties of this graph has been studied. In this paper, we take a combinatorial approach to investigate the rainbow vertex connectivity of the small-world Farey graph. A graph is rainbow vertex-connected when there exists a path, in which all of its internal vertices have distinct colors, between each pair of vertices. We give a coloring so that the graph is rainbow vertex-connected. The rainbow vertex-connection number of the small-world Farey graph, which is the minimum number of colors needed so that the graph is rainbow vertex-connected, is one less than the diameter of the graph.
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