Safe 3-coloring of graphs

TL;DR

This paper introduces the concept of safe 3-coloring in graphs, motivated by secret sharing applications, and analyzes which graphs can be safely colored with three colors to prevent information breaches.

Contribution

It defines the new concept of safe 3-coloring in graphs and investigates the classes of graphs that admit such colorings.

Findings

Identifies conditions for safe 3-colorings in various graphs

Provides characterization of graphs with safe 3-colorings

Connects graph coloring to secure secret sharing methods

Abstract

The applications of graph coloring are diverse and many so lots of new types of coloring are being proposed and explored. Here we define a safe k-coloring, motivated by the application of coloring to secret sharing. Secret sharing is a way of securing a secret from a number of attackers by dividing it into parts and then distributing those parts to some persons, represented here by graph vertices. Parts of the secret are represented by colors which are then assigned to the vertices under certain conditions, making a coloring safe if a predetermined number of attackers cannot read the whole secret, nor disable the rest of the group from doing so. We observe a fixed number of colors, namely 3, and analyze what kind of graphs have a safe 3-coloring.