Hamiltonicity of planar graphs with a forbidden minor

TL;DR

This paper proves that 3-connected planar graphs without a K_{2,5} minor are Hamiltonian, extending known results for 4-connected graphs and exploring limitations with specific minors.

Contribution

It establishes Hamiltonicity for K_{2,5}-minor-free 3-connected planar graphs, a new class beyond previously known 4-connected cases.

Findings

K_{2,5}-minor-free 3-connected planar graphs are Hamiltonian

The result does not extend to general 3-connected planar graphs

Counterexamples exist for K_{2,6}-minor-free 3-connected planar graphs

Abstract

Tutte showed that $4$-connected planar graphs are Hamiltonian, but it is well known that $3$-connected planar graphs need not be Hamiltonian. We show that $K_{2,5}$-minor-free $3$-connected planar graphs are Hamiltonian. This does not extend to $K_{2,5}$-minor-free $3$-connected graphs in general, as shown by the Petersen graph, and does not extend to $K_{2,6}$-minor-free $3$-connected planar graphs, as we show by an infinite family of examples.