Abstract
This paper aims to survey Hamiltonicity of graphs on surfaces, including stronger (e.g. Hamiltonian-connectedness) and weaker (e.g. containing Hamiltonian paths and spanning trees with certain conditions) properties. Toughness and scattering number conditions are necessary conditions for graphs to have such properties. Since every k-connected graph on a surface \(F^2\) satisfies some toughness and scattering number condition, we can expect that “every k-connected graph on a surface \(F^2\) satisfies the property \({ \mathcal {P}}\)”. We explain which triple \((k, F^2, { \mathcal {P}})\) makes the statement true from the viewpoint of toughness and scattering number of graphs.
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