New Techniques Based On Odd-Edge Total Colorings In Topological Cryptosystem

TL;DR

This paper introduces new odd-edge total coloring techniques for topological cryptography, utilizing complex graph constructions and leaf-adding algorithms to develop twin-graphic lattices with potential quantum computing applications.

Contribution

It proposes four novel odd-magic-type colorings and algorithms for generating graphs with these colorings, advancing the theory of topological cryptography and graphic lattices.

Findings

Development of four new odd-magic-type colorings.

Construction of complex caterpillar-graphic and complementary lattices.

Establishment of twin-type $W$-magic graphic lattices and homomorphisms.

Abstract

For building up twin-graphic lattices towards topological cryptograph, we define four kinds of new odd-magic-type colorings: odd-edge graceful-difference total coloring, odd-edge edge-difference total coloring, odd-edge edge-magic total coloring, and odd-edge felicitous-difference total coloring in this article. Our RANDOMLY-LEAF-ADDING algorithms are based on adding randomly leaves to graphs for producing continuously graphs admitting our new odd-magic-type colorings. We use complex graphs to make caterpillar-graphic lattices and complementary graphic lattices, such that each graph in these new graphic lattices admits a uniformly $W$-magic total coloring. On the other hands, finding some connections between graphic lattices and integer lattices is an interesting research, also, is important for application in the age of quantum computer. We set up twin-type $W$-magic graphic lattices (as public graphic lattices vs private graphic lattices) and $W$-magic graphic-lattice homomorphism for producing more complex topological number-based strings.