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Grinberg’s criterion. (English) Zbl 1400.05138
Summary: We generalize Grinberg’s Hamiltonicity criterion for planar graphs. To this end, we first prove a technical theorem for embedded graphs. As a special case of a corollary of this theorem we obtain Zaks’ extension of Grinberg’s criterion (which encompasses earlier work of K. R. Gehner [Networks 6, 131–138 (1976; Zbl 0354.05039)] and Y. Shimamoto [J. Comb. Theory, Ser. B 24, 169–180 (1978; Zbl 0395.05052)]), but the result also implies Grinberg’s formula in its original form in a much broader context. Further implications are a short proof for a slightly strengthened criterion of Lewis bounding the length of a shortest closed walk from below as well as a generalization of a theorem due to J. A. Bondy and R. Häggkvist [Aequationes Math. 22, 42–45 (1981; Zbl 0464.05037)].
MSC:
05C45 Eulerian and Hamiltonian graphs
05C10 Planar graphs; geometric and topological aspects of graph theory
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