A '1+1' Algorithm for the Hamilton Cycle Problem

TL;DR

This paper introduces the '1+1' algorithm, an efficient method with polynomial worst-case complexity for deciding whether a graph contains a Hamilton cycle, advancing the understanding of Hamilton graph characterization.

Contribution

It develops a new algorithm based on improved Grinberg Theorem properties, providing a more precise and tractable solution for the Hamilton cycle problem.

Findings

Algorithm terminates in $O(|E(G)|^3)$ worst time

Provides a necessary condition for Hamilton graphs

Advances theoretical understanding of Hamilton graph characterization

Abstract

Deciding if a graph is a Hamilton graph, also named the Hamilton cycle problem, is important for discrete mathematics and computer science. Due to no characterization to identify Hamilton graphs effectively, there are no tractable algorithms to solve the Hamilton cycle problem. Grinberg Theorem is a necessary condition only for planar Hamilton graphs. In this paper, based on new studies on the Grinberg Theorem, in which we provided new properties of Hamilton graphs with respect to the cycle bases and improved the Grinberg Theorem to derive an efficient condition for Hamilton graphs, we present a new precise algorithm for deciding Hamilton graphs, named the '1+1' algorithm. Theoretically, the '1+1' algorithm terminates in $O(|E(G)|^3)$ worst time complexity, where $|\textit{E}(\textit{G})|$ is the size of the given graph $\textit{G}$.