Chromatic Number of Graphs with Special Distance Sets-V

Venkataraman Yegnanarayanan,Angamuthu Parthiban

doi:10.4236/ojdm.2013.31001

Abstract

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. It is known that x(G(Z,D) )=4 where P is a set of Prime numbers. So we can allocate the subsets D of P to four classes, accordingly as is 1 or 2 or 3 or 4. In this paper we have considered the open problem of characterizing class three and class four sets when the distance set D is not only a subset of primes P but also a special class of primes like Additive primes, Deletable primes, Wedderburn-Etherington Number primes, Euclid-Mullin sequence primes, Motzkin primes, Catalan primes, Schroder primes, Non-generous primes, Pell primes, Primeval primes, Primes of Binary Quadratic Form, Smarandache-Wellin primes, and Highly Cototient number primes. We also have indicated the membership of a number of special classes of prime numbers in class 2 category.

Highlights

The graphs considered in this paper are simple and undirected
We study here the open problem of characterizing the class 4 sets mentioned in the problem for the prime distance graph whose vertex set is Z and the distance set D is a subset of primes P but are they are a special set of primes
Let A denote the set of any one of these special class of prime numbers listed in the theorem

Summary

Introduction

The graphs considered in this paper are simple and undirected. A k -coloring of a graph G is an assignment of k different colors to the vertices of G such that adjacent vertices receive different colors. Of V1 with color 1 and the vertices of V2 with color 2 Erdos has mentioned this problem as one of his favorite problems. Theorem 1.1 ([2]) Let k be a positive integer, and let the graph G have the property that any finite subgraph is k -colorable. Theorem 1.1 allows us to look for the maximum number of colors needed for the finite subsets. Despite the age of this problem, very little progress has been made since the initial bounds on were discovered shortly after the problem’s creation This fact is a testament to the difficulty of the problem and in the absence of progress on the main problem, a number of restricted versions and related questions have been studied. The authors have already made some progress in [5,6,7,8,9,10,11]

Prime Distance Graph

Certain Known Results

Some Special Set of Prime Numbers

Chromatic Number of Certain Prime Distance Graphs

Conclusion