Finite Prime Distance Graphs and 2-Odd Graphs

TL;DR

This paper explores prime distance and 2-odd graphs, establishing which classes of graphs fit these definitions and linking some cases to famous conjectures like the Twin Prime Conjecture.

Contribution

It characterizes prime distance and 2-odd graphs, proves several classes are such graphs, and connects certain cases to major unresolved conjectures.

Findings

Trees, cycles, and bipartite graphs are prime distance graphs.

Dutch windmill and paper mill graphs relate to twin primes and dePolignac's conjecture.

Characterization of 2-odd graphs via edge colorings.

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 dePolignac'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.