[Paper Review] On planar regular graphs degree three without Hamiltonian cycles

Emanuels Grinbergs|ArXiv.org|Aug 18, 2009

Advanced Graph Theory Research1 references3 citations

TL;DR

This paper presents a necessary condition for the existence of Hamiltonian cycles in planar 3-regular graphs and constructs explicit examples of such graphs that are non-Hamiltonian. Based on Grinberg's theorem, it demonstrates that certain structural constraints in planar graphs prevent Hamiltonian cycles, providing foundational counterexamples in graph theory.

ABSTRACT

Necessary condition to have Hamiltonian cycle in planar graph is given. Examples of regular planar graphs degree three without Hamiltonian cycle are built.

Motivation & Objective

• To establish a necessary condition for the existence of Hamiltonian cycles in planar 3-regular graphs.
• To investigate structural properties that prevent Hamiltonian cycles in such graphs.
• To construct explicit examples of planar 3-regular graphs that lack Hamiltonian cycles.
• To formalize and disseminate Grinberg's earlier unpublished work on non-Hamiltonian 3-regular planar graphs.
• To provide a theoretical foundation for understanding the limitations of Hamiltonian cycles in restricted planar graph classes.

Proposed method

• Derives a necessary condition for Hamiltonian cycles in planar 3-regular graphs based on face cycle decompositions.
• Applies Grinberg's theorem, which relates the number of vertices in even- and odd-length cycles of a planar graph.
• Uses combinatorial construction techniques to generate planar 3-regular graphs with specific face-length distributions.
• Employs graph duality and cycle space analysis to verify non-Hamiltonicity.
• Validates results through structural analysis of face cycles and their parity constraints.
• Translates and publishes the original 1968 Russian work of Emanuels Grinbergs for broader accessibility.

Experimental results

Research questions

• RQ1 What necessary condition must a planar 3-regular graph satisfy to admit a Hamiltonian cycle?
• RQ2 Can planar 3-regular graphs exist that are structurally incapable of supporting a Hamiltonian cycle?
• RQ3 What specific graph constructions demonstrate the failure of Hamiltonicity under Grinberg’s condition?
• RQ4 How do face-length distributions influence the existence of Hamiltonian cycles in 3-regular planar graphs?
• RQ5 What is the role of cycle parity in determining Hamiltonian properties of planar graphs?

Key findings

• A necessary condition for Hamiltonian cycles in planar 3-regular graphs is derived using Grinberg’s theorem.
• Explicit examples of planar 3-regular graphs without Hamiltonian cycles are constructed.
• The construction relies on violating the parity condition in Grinberg’s formula, ensuring non-Hamiltonicity.
• The paper confirms that not all planar 3-regular graphs are Hamiltonian, even under strong regularity and planarity constraints.
• The original 1968 Russian work by Grinbergs is validated and made accessible through this English translation.
• The results establish a foundational obstruction to Hamiltonicity in a well-studied class of graphs.