Abstract. In this paper first, we give a brief introduction about integer distance graphs. An integer distance graph is a graph with the set of integers as vertex set and an edge joining two vertices and if and only if where is a subset of the positive integers. If is a subset of then we call a prime distance graph. Second, we obtain a partial solution to a general open problem of characterizing a class of prime distance graphs. Third, we compute the vertex arboricity of certain prime distance graphs. Fourth, we give a brief review regarding circulant graphs and highlight its importance in the computation of chromatic number of distance graphs with appropriate references. Fifth, we introduce the notion of pseudochromatic coloring and obtain certain results concerning circulant graphs and distance graphs.

Received: April 1, 2013

AMS Subject Classification: 05C15

Key Words and Phrases: chromatic number, circulant graphs, coloring, prime distance graphs

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