Elsevier

Discrete Mathematics

Volume 313, Issue 20, 28 October 2013, Pages 2281-2291
Discrete Mathematics

Finite prime distance graphs and 2-odd graphs

https://doi.org/10.1016/j.disc.2013.06.005Get rights and content
Under an Elsevier user license
open archive

Abstract

A graph G is a prime distance graph (respectively, a 2-odd graph) if its vertices can be labeled with distinct integers such that for any two adjacent vertices, the difference of their labels is prime (either 2 or odd). We prove that trees, cycles, and bipartite graphs are prime distance graphs, and that Dutch windmill graphs and paper mill graphs are prime distance graphs if and only if the Twin Prime Conjecture and de Polignac’s Conjecture are true, respectively. We give a characterization of 2-odd graphs in terms of edge colorings, and we use this characterization to determine which circulant graphs of the form Circ(n,{1,k}) are 2-odd and to prove results on circulant prime distance graphs.

Keywords

Distance graphs
Prime distance graphs
Difference graphs

Cited by (0)