Vertex Coloring of Certain Distance Graphs

Abstract

In this paper first, we give a brief introduction about integer distance graphs. An integer distance graph is a graph G(Z, D) with the set of integers as vertex set and an edge joining two vertices u and v if and only if |u − v| ∈ D where D is a subset of the positive integers. If D is a subset of P then we call G(Z, D) 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.

Some Results on Prime and Distinct Prime Distance Labeling of Graphs

“…The integer distance graph is the graph with vertex set (the set of all integers) and two vertices and are adjacent if and only if Then the prime distance graph is the distance graph with , the set of all primes. For a detailed study on integer distance and prime distance graphs, see [13][14][15].…”