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In this paper, we study the large‐scale structure of dense regular graphs. This involves the notion of robust expansion, a recent concept which has already been used successfully to settle several longstanding problems. Roughly speaking, a graph is robustly expanding if it still expands after the deletion of a small fraction of its vertices and edges. Our main result allows us to harness the useful consequences of robust expansion even if the graph itself is not a robust expander. It states that every dense regular graph can be partitioned into ‘robust components’, each of which is a robust expander or a bipartite robust expander. We apply our result to obtain (amongst others) the following.   We prove that whenever    ε  >  0   , every sufficiently large   3  ‐connected   D  ‐regular graph on   n   vertices with    D  ⩾  (    1  4    +  ε  )  n    is Hamiltonian. This asymptotically confirms the only remaining case of a conjecture raised independently by Bollobás and Häggkvist in the 1970s.    We prove an asymptotically best possible result on the circumference of dense regular graphs of given connectivity. The   2  ‐connected case of this was conjectured by Bondy and proved by Wei.

The robust component structure of dense regular graphs and applications