Some remarks on Jaeger's dual-hamiltonian conjecture

Bill Jackson; Carol A. Whitehead

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Barnette conjectured that each planar, bipartite, cubic, and 3-connected graph is hamiltonian. We prove that this conjecture is equivalent to the statement that there is a constant c > 0 such that each graph G of this class contains a path on at least c|V (G)| vertices.

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Let $G$ be a finite connected graph with minimum degree $δ$ . The leaf number $L (G)$ of $G$ is defined as the maximum number of leaf vertices contained in a spanning tree of $G$ . We prove that if $δ \geq \frac{1}{2} (L (G) + 1)$ , then $G$ is 2-connected. Further, we deduce, for graphs of girth greater than 4, that if $δ \geq \frac{1}{2} (L (G) + 1)$ , then $G$ contains a spanning path. This provides a partial solution to a conjecture of the computer program Graffiti.pc [DeLaVi na and Waller, Spanning trees with many leaves and average distance, Electron. J. Combin....