On Sufficient Conditions for Hamiltonicity in Dense Graphs

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. This solves the embedding problem that underlies multiple lines of research on sufficient conditions for Hamiltonicity. As an application, we recover several old and new results, and prove versions of the Bandwidth Theorem under Ore-type degree conditions, Pósa-type degree conditions, deficiency-type conditions and for balanced partite graphs.