Chromatic number of graphs with special distance sets, I

Venkataraman Yegnanarayanan

Algebra and discrete mathematics, Tome 17 (2014) no. 1, pp. 135-160

Résumé

Given a subset $D$ of positive integers, an integer distance graph is a graph $G(\mathbb{Z}, D)$ with the set $\mathbb{Z}$ of integers as vertex set and with an edge joining two vertices $u$ and $v$ if and only if $|u - v| \in D$. In this paper we consider the problem of determining the chromatic number of certain integer distance graphs $G(\mathbb{Z}, D)$whose distance set $D$ is either 1) a set of $(n+1)$ positive integers for which the $n^{th}$ power of the last is the sum of the $n^{th}$ powers of the previous terms, or 2) a set of pythagorean quadruples, or 3) a set of pythagorean $n$-tuples, or 4) a set of square distances, or 5) a set of abundant numbers or deficient numbers or carmichael numbers, or 6) a set of polytopic numbers, or 7) a set of happy numbers or lucky numbers, or 8) a set of Lucas numbers, or 9) a set of $\mathcal{U}$lam numbers, or 10) a set of weird numbers. Besides finding the chromatic number of a few specific distance graphs we also give useful upper and lower bounds for general cases. Further, we raise some open problems.

Détail

Keywords: chromatic number, prime distance graph, unit distance graph.