On sufficient conditions for spanning structures in dense graphs

TL;DR

This paper establishes new sufficient conditions in dense graphs that guarantee the existence of spanning structures like Hamilton cycles and their powers, extending previous results and solving key embedding problems.

Contribution

It generalizes the Robust Expander Theorem to broader classes of spanning structures and addresses a major open question in dense graph theory.

Findings

Generalized conditions for Hamiltonicity and cycle powers in dense graphs

Recovered and extended Bandwidth Theorems under various degree conditions

Solved embedding problems for spanning structures in diverse graph classes

Abstract

We study structural conditions in dense graphs that guarantee the existence of vertex-spanning substructures such as Hamilton cycles. It is easy to see that every Hamiltonian graph is connected, has a perfect fractional matching and, excluding the bipartite case, contains an odd cycle. A simple consequence of the Robust Expander Theorem of K\"uhn, Osthus and Treglown tells us that any large enough graph that robustly satisfies these properties must already be Hamiltonian. Our main result generalises this phenomenon to powers of cycles and graphs of sublinear bandwidth subject to natural generalisations of connectivity, matchings and odd cycles. This answers a question of Ebsen, Maesaka, Reiher, Schacht and Sch\"ulke and solves the embedding problem that underlies multiple lines of research on sufficient conditions for spanning structures in dense graphs. As applications, we recover and establish Bandwidth Theorems in a variety of settings including Ore-type degree conditions, P\'osa-type degree conditions, deficiency-type conditions, locally dense and inseparable graphs, multipartite graphs as well as robust expanders.