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Hamiltonicity of cubic 3-connected k -Halin graphs
Abstract We investigate here how far we can extend the notion of a Halin graph such that hamiltonicity is preserved. Let H = T βͺ C be a Halin graph, T being a tree and C the outer cycle. A k-Halin graph G can be obtained from H by adding edges while keeping planarity, joining vertices of H β C, such that G β C has at most k cycles. We prove that, in the class of cubic 3-connected graphs, all 14-Halin graphs are hamiltonian and all 7-Halin graphs are 1-edge hamiltonian. These results are best possible