Proof of Komlós's conjecture on Hamiltonian subsets

Abstract

Abstract. Komlós conjectured in 1981 that among all graphs with minimum degree at least d, the complete graph K d+1 minimises the number of Hamiltonian subsets, where a subset of vertices is Hamiltonian if it contains a spanning cycle. We prove this conjecture when d is sufficiently large. In fact we prove a stronger result: for large d, any graph G with average degree at least d contains almost twice as many Hamiltonian subsets as K d+1 , unless G is isomorphic to K d+1 or a certain other graph which we specify.

Extremal Density for Sparse Minors and Subdivisions

“…Our proof utilises both pseudorandomness from Szemerédi's regularity lemma and expansions for sparse graphs. The particular expander that we shall make use of is an extension of the one introduced by Komlós and Szemerédi, which has played an important role in some recent developments on sparse graph embedding problems, see for example, [14,21].…”

“…Hence, we have (5). To find such webs, we follow the strategy of [14,Lemma 5.7]. We include the proof in the online appendix [12].…”

Towards the Erdős-Gallai cycle decomposition conjecture

“…Sublinear expansion is a weaker notion of this classical expansion introduced by Komlós and Szemerédi [34,35], where we take a much smaller value of λ, but which is significant as every graph contains a sublinear expander H with λ = Θ(1/ log 2 |H|) (and even has a nice decomposition into sublinear expanders, as we will prove and use). Komlós and Szemerédi used sublinear expansion to find minors in sparse graphs, and more recently sublinear expansion has found a host of other applications (see, for example, [8,21,22,26,27,33,39,40,41,42,45,50]).…”

Immersion of complete digraphs in Eulerian digraphs

“…Our proof consists of three key steps. First, we use a notion of sublinear expansion introduced by Komlós and Szemerédi [11,12], which played a key role in recent progress on several long-standing conjectures (see, e.g., [7,8,10,14,15]). Our proof is somewhat unusual in that it applies this notion of expansion to digraphs.…”