English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

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