On graph coloring check-digit method

TL;DR

This paper introduces a method to convert graphs to binary numbers and vice versa, derives bounds on graph colorability, and proposes a graph coloring-based check-digit scheme with improved error detection capabilities.

Contribution

It presents a novel conversion method between graphs and binary numbers, derives bounds on graph colorability, and develops a new check-digit scheme based on graph coloring.

Findings

Feasibility of the check-digit scheme increases with number size.

Derived upper bounds for the number of graphs with fixed vertices and colors.

Proposed scheme enhances error detection probability.

Abstract

We show a method how to convert any graph into the binary number and vice versa. We derive upper bound for maximum number of graphs, that, have fixed number of vertices and can be colored with n colors (n is any given number). Proof for the result is outlined. Next, graph coloring based check-digit scheme is proposed. We use quantitative result derived, to show, that feasibility of the proposed scheme increases with size of the number which digits are checked, and overall probability of digits errors.