Robust expansion and hamiltonicity

This thesis contains four results in extremal graph theory relating to the recent notion of robust expansion, and the classical notion of Hamiltonicity. In Chapter 2 we prove that every sufficiently large ‘robustly expanding’ digraph which is dense and regular has an approximate Hamilton decomposition. This provides a common generalisation of several previous results and in turn was a crucial tool in Kühn and Osthus’s proof that in fact these conditions guarantee a Hamilton decomposition, thereby proving a conjecture of Kelly from 1968 on regular tournaments. In Chapters 3 and 4, we prove that every sufficiently large 3-connected $D$-regular graph on $n$ vertices with $D$ ≥ n/4 contains a Hamilton cycle. This answers a problem of Bollobás and Häggkvist from the 1970s. Along the way, we prove a general result about the structure of dense regular graphs, and consider other applications of this. Chapter 5 is devoted to a degree sequence analogue of the famous Pósa conjecture. Our main result is the following: if the $i$$^{th}$ largest degree in a sufficiently large graph $G$ on n vertices is at least a little larger than $n$/3 + $i$ for $i$ ≤ $n$/3, then $G$ contains the square of a Hamilton cycle

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum

Large bichromatic point sets admit empty monochromatic 4-gons

We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).Postprint (published version

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum