Abstract

We study structural conditions in dense graphs that guarantee the existence of vertex-spanning substructures such as Hamilton cycles. Recall that every Hamiltonian graph is connected, has an almost perfect matching and, excluding the bipartite case, contains an odd cycle. Our main result states that any large enough graph that robustly satisfies these properties must already be Hamiltonian. Moreover, the same holds for powers of cycles and the bandwidth setting subject to natural generalizations of connectivity, matchings and odd cycles.

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