Coloring Distance Graphs on the Integers

Glenn G. Chappell

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arXiv · math/9805084

Coloring Distance Graphs on the Integers

Glenn G. Chappell

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Abstract

Given a set D of positive integers, the associated distance graph on the integers is the graph with the integers as vertices and an edge between distinct vertices if their difference lies in D. We investigate the chromatic numbers of distance graphs. We show that, if $D = {d_1,d_2,d_3,...}$, with $d_n | d_{n+1}$ for all n, then the distance graph has a proper 4-coloring. We further find the exact chromatic numbers of all such distance graphs. Next, we characterize those distance graphs that have periodic proper colorings and show a relationship between the chromatic number and the existence of periodic proper colorings.

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Glenn G. Chappell. 1998-05-19. Coloring Distance Graphs on the Integers. https://arxiv.org/abs/math/9805084

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