Abstract
An edge of a 3-connected graph is contractible if its contraction results in a graph which is still 3-connected. All 3-connected graphs with seven or more vertices are known to have at least three contractible edges on any longest cycle. Recently, it has been conjectured that any non-Hamiltonian 3-connected graph has at least six contractible edges on any longest cycle. We prove this conjecture and provide a construction to show that it is best possible.
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