A note on the Grinberg condition in the cycle spaces

TL;DR

This paper offers a new combinatorial interpretation of Grinberg's condition in cycle spaces, explaining its limitations and aiming to improve criteria for identifying Hamilton graphs.

Contribution

It introduces a novel interpretation of Grinberg's condition using cycle space bases and the Inclusion-Exclusion Principle, clarifying its insufficiency for Hamiltonicity.

Findings

Derived an equality related to inside faces using cycle space basis

Provided a new combinatorial perspective on Grinberg's condition

Explained why Grinberg's theorem is not sufficient for Hamilton graphs

Abstract

Finding a Hamilton graph from simple connected graphs is an important problem in discrete mathematics and computer science. Grinberg Theorem is a well-known necessary condition for planar Hamilton graphs. It divides a plane into two parts: inside and outside faces. The sum of inside faces in a Hamilton graph is a Hamilton cycle. In this paper, using a basis of the cycle space to represent a graph and by the Inclusion-Exclusion Principle, we derive the equality with respect to the inside faces that can be also obtained from Grinberg Theorem. By further investigating the cycle structure of inside faces, we give a new combinatorial interpretation to Grinberg's condition, which explains why Grinberg Theorem is not sufficient for Hamilton graphs. Our results will improve deriving an efficient condition for Hamilton graphs.