Solutions of Grinberg equation and removable cycles in a cycle basis

TL;DR

This paper explores the use of cycle bases in graphs to analyze Grinberg's theorem, aiming to clarify its role as a necessary condition for Hamiltonicity and extend its application to simple graphs.

Contribution

It introduces a new approach using cycle bases to reformulate Grinberg's theorem, providing insights into its necessity and sufficiency for Hamiltonian cycles in graphs.

Findings

Reformulation of Grinberg theorem using cycle bases

Clarification of Grinberg theorem as a necessary condition

Application of the theorem to simple graphs

Abstract

Let G (V, E) be a simple graph with vertex set V and edge set E. A generalized cycle is a subgraph such that any vertex degree is even. A simple cycle (briefly in a cycle) is a connected subgraph such that every vertex has degree 2. A basis of the cycle space is called a cycle basis of G (V, E). A cycle basis where the sum of the weights of the cycles is minimal is called a minimum cycle basis of G. Grinberg theorem is a necessary condition to have a Hamilton cycle in planar graphs. In this paper, we use the cycles of a cycle basis to replace the faces and obtain an equality of inner faces in Grinberg theorem, called Grinberg equation. We explain why Grinberg theorem can only be a necessary condition of Hamilton graphs and apply the theorem, to be a necessary and sufficient condition, to simple graphs.