Multicoloring of Graphs to Secure a Secret

TL;DR

This paper introduces a new vertex multicoloring scheme called highly a-resistant vertex k-multicoloring, designed to secure secrets against attackers, and determines minimal graph sizes and color counts for small attacker numbers.

Contribution

It proposes a novel multicoloring method inspired by secret sharing security needs and analyzes minimal graph and color requirements for small attacker scenarios.

Findings

Determines minimal graph size for small attacker numbers.

Establishes minimal number of colors needed.

Introduces a new multicoloring scheme for secret security.

Abstract

Vertex coloring and multicoloring of graphs are a well known subject in graph theory, as well as their applications. In vertex multicoloring, each vertex is assigned some subset of a given set of colors. Here we propose a new kind of vertex multicoloring, motivated by the situation of sharing a secret and securing it from the actions of some number of attackers. We name the multicoloring a highly $a$-resistant vertex $k$-multicoloring, where $a$ is the number of the attackers, and $k$ the number of colors. For small values $a$ we determine what is the minimal number of vertices a graph must have in order to allow such a coloring, and what is the minimal number of colors needed.