Distance Graphs of Metric Spaces with Rosenbloom - Tsfasman metric

TL;DR

This paper explores the properties of distance graphs in metric spaces equipped with the Rosenbloom-Tsfasman metric, including vertex degrees, components, and chromatic numbers, extending understanding beyond the traditional Hamming metric.

Contribution

It provides a detailed analysis of the structure of distance graphs under the Rosenbloom-Tsfasman metric, a generalization of the Hamming metric, including graph properties and coloring.

Findings

Determined the degrees of vertices in the graphs.

Described the components of the graphs.

Calculated the chromatic number of the graphs.

Abstract

Rosenbloom and Tsfasman introduced a new metric (RT metric) which is a generalization of the Hamming metric. In this paper we study the distance graphs of spaces $Z_q^n$ and $S_n$ with Rosenbloom -Tsfasman metric. We also describe the degrees of vertices, components and the chromatic number of these graphs.