Graph Homomorphisms Based On Particular Total Colorings of Graphs and Graphic Lattices

TL;DR

This paper introduces new types of graph homomorphisms based on total colorings and graphic lattices, linking graph theory with cryptography and proposing problems like Number String Decomposition.

Contribution

It combines graph homomorphisms with total colorings to create novel homomorphism types and lattice structures, expanding theoretical understanding.

Findings

Defined totally-colored graph homomorphisms and graphic-lattice homomorphisms.

Established new problems such as Number String Decomposition.

Connected graph homomorphisms to cryptographic concepts.

Abstract

Lattice-based cryptography is not only for thwarting future quantum computers, and is also the basis of Fully Homomorphic Encryption. Motivated from the advantage of graph homomorphisms we combine graph homomorphisms with graph total colorings together for designing new types of graph homomorphisms: totally-colored graph homomorphisms, graphic-lattice homomorphisms from sets to sets, every-zero graphic group homomorphisms from sets to sets. Our graph-homomorphism lattices are made up by graph homomorphisms. These new homomorphisms induce some problems of graph theory, for example, Number String Decomposition and Graph Homomorphism Problem.