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Every 2-connected, cubic, planar graph with faces of size at most 6 is Hamiltonian

Sihong Shao, Yuxuan Wu

TL;DR

The paper addresses the Barnette-Goodey conjecture under the weakest natural connectivity constraint by proving that every $2$-connected, planar, cubic graph with all faces of size at most $6$ is Hamiltonian. The authors develop a structural framework based on planar embeddings, outer-face analysis, and $2$-edge-cuts, and then apply induction on the vertex count to reduce to smaller graphs whose Hamiltonicity can be established and stitched together into a Hamiltonian cycle for the original graph. A key contribution is showing the $6$-face bound is tight, via a counterexample with faces up to size $7$, and thus identifying the precise boundary where the conjecture holds under the given connectivity. The work situates this result within the broader landscape of Barnette’s conjecture, notes that further weakening rapidly leads to NP-hardness, and highlights the use of planar-embedding techniques in proving Hamiltonicity.

Abstract

We prove that every 2-connected, cubic, planar graph with faces of size at most 6 is Hamiltonian, and show that the 6-face condition is tight. Our results push the connectivity condition of the Barnette-Goodey conjecture to the weakest possible.

Every 2-connected, cubic, planar graph with faces of size at most 6 is Hamiltonian

TL;DR

The paper addresses the Barnette-Goodey conjecture under the weakest natural connectivity constraint by proving that every -connected, planar, cubic graph with all faces of size at most is Hamiltonian. The authors develop a structural framework based on planar embeddings, outer-face analysis, and -edge-cuts, and then apply induction on the vertex count to reduce to smaller graphs whose Hamiltonicity can be established and stitched together into a Hamiltonian cycle for the original graph. A key contribution is showing the -face bound is tight, via a counterexample with faces up to size , and thus identifying the precise boundary where the conjecture holds under the given connectivity. The work situates this result within the broader landscape of Barnette’s conjecture, notes that further weakening rapidly leads to NP-hardness, and highlights the use of planar-embedding techniques in proving Hamiltonicity.

Abstract

We prove that every 2-connected, cubic, planar graph with faces of size at most 6 is Hamiltonian, and show that the 6-face condition is tight. Our results push the connectivity condition of the Barnette-Goodey conjecture to the weakest possible.
Paper Structure (3 sections, 8 theorems, 9 figures)

This paper contains 3 sections, 8 theorems, 9 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and lemmas
  3. Proof of the main result

Key Result

Theorem 1

Every 2-connected, planar, cubic graph with faces of size at most 6 is Hamiltonian.

Figures (9)

  • Figure 1: The 6-face condition of Theorem \ref{['t1']} is tight: An example. The larger Graph $G$ in the left is a 2-connected, planar, non-Hamiltonian, cubic graph with faces of size at most 7. To see $G$ is indeed non-Hamiltonian, it suffices to use the fact that every Hamiltonian cycle passing through the structure in the right must use the edge $e$. If $G$ is Hamiltonian, then its Hamiltonian cycle must use the edges $1$, $2$, and $3$ in the left plot, which results in a contradiction.
  • Figure 2: The proof of Corollary \ref{['c1']}.
  • Figure 3: The proof of Lemma \ref{['l51']}.
  • Figure 4: The proof of Lemma \ref{['l52']}.
  • Figure 5: The proof of Lemma \ref{['l53']}: The smallest graph $G$ and its Hamiltonian path marked with double dash.
  • ...and 4 more figures

Theorems & Definitions (13)

  • Theorem 1
  • Proposition 2
  • Corollary 3
  • proof
  • Proposition 4
  • Proposition 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • ...and 3 more