Evolutionary Hamiltonian Graph Theory

TL;DR

This paper introduces Large Cycles Theory (LCT), a rapidly evolving mathematical framework inspired by evolutionary principles, which unifies many NP-complete cycle problems in graph theory and offers insights into biological evolution mechanisms.

Contribution

It presents LCT as a novel, rapidly evolving domain that models the evolution of theorems in graph theory, drawing parallels with biological evolution and expanding understanding of NP-complete problems.

Findings

LCT theorems descend from primitive propositions.

LCT evolves faster than biological evolution.

Theories in LCT form a diverse 'species' of mathematical results.

Abstract

We present an alternative domain concerning mathematics to investigate universal evolution mechanisms by focusing on large cycles theory (LCT) - a simplified version of well-known hamiltonian graph theory. LCT joins together a number of $NP$-complete cycle problems in graph theory. $NP$-completeness is the kay factor insuring (by conjecture of Cook) the generation of endless developments and great diversity around large cycles problems. Originated about 60 years ago, the individuals (claims, propositions, lemmas, conjectures, theorems, and so on) in LCT continually evolve and adapt to their environment by an iterative process from primitive beginnings to best possible theorems based on inductive reasoning. LCT evolves much more rapidly than biosphere and has a few thousand pronounced species (theorems). Recall that life on earth with more than 2 million species was originated about 3.7 billion years ago and evolves extremely slowly. We show that all theorems in LCT have descended from some common primitive propositions such as "every complete graph is hamiltonian" or "every graph contains a cycle of length at least one" via improvements, modifications and three kinds of generalizations - closing, associating and extending. It is reasonable to review Darwinian mechanisms in light of LCT evolution mechanisms (especially inductive reasoning) including the origin and macroevolution disputable phenomena in the biosphere.