14 research outputs found
Small Complete Minors Above the Extremal Edge Density
A fundamental result of Mader from 1972 asserts that a graph of high average
degree contains a highly connected subgraph with roughly the same average
degree. We prove a lemma showing that one can strengthen Mader's result by
replacing the notion of high connectivity by the notion of vertex expansion.
Another well known result in graph theory states that for every integer t
there is a smallest real c(t) so that every n-vertex graph with c(t)n edges
contains a K_t-minor. Fiorini, Joret, Theis and Wood conjectured that if an
n-vertex graph G has (c(t)+\epsilon)n edges then G contains a K_t-minor of
order at most C(\epsilon)log n. We use our extension of Mader's theorem to
prove that such a graph G must contain a K_t-minor of order at most
C(\epsilon)log n loglog n. Known constructions of graphs with high girth show
that this result is tight up to the loglog n factor
Polynomial expansion and sublinear separators
Let be a class of graphs that is closed under taking subgraphs.
We prove that if for some fixed , every -vertex graph of
has a balanced separator of order , then any
depth- minor (i.e. minor obtained by contracting disjoint subgraphs of
radius at most ) of a graph in has average degree . This confirms a conjecture of Dvo\v{r}\'ak
and Norin.Comment: 6 pages, no figur
A tight Erd\H{o}s-P\'osa function for wheel minors
Let denote the wheel on vertices. We prove that for every integer
there is a constant such that for every integer
and every graph , either has vertex-disjoint subgraphs each
containing as minor, or there is a subset of at most
vertices such that has no minor. This is best possible, up to the
value of . We conjecture that the result remains true more generally if we
replace with any fixed planar graph .Comment: 15 pages, 1 figur
Hypergraphs with no tight cycles
We show that every r-uniform hypergraph on n vertices which does not contain a tight cycle has has at most O(n^{r-1}(log n)^{5}) edges. This is an improvement on the previously best-known bound, of n^{r-1}e^{O(\sqrt{log n})} due to Sudakov and Tomon, and our proof builds on their work. A recent
construction of B. Janzer implies that our bound is tight up to an O((log n)^{4} log log n) factor
Logarithmically-small Minors and Topological Minors
Mader proved that for every integer there is a smallest real number
such that any graph with average degree at least must contain a
-minor. Fiorini, Joret, Theis and Wood conjectured that any graph with
vertices and average degree at least must contain a -minor
consisting of at most vertices. Shapira and Sudakov
subsequently proved that such a graph contains a -minor consisting of at
most vertices. Here we build on their method
using graph expansion to remove the factor and prove the
conjecture.
Mader also proved that for every integer there is a smallest real number
such that any graph with average degree larger than must contain
a -topological minor. We prove that, for sufficiently large , graphs
with average degree at least contain a -topological
minor consisting of at most vertices. Finally, we show
that, for sufficiently large , graphs with average degree at least
contain either a -minor consisting of at most
vertices or a -topological minor consisting of at most
vertices.Comment: 19 page
Rainbow Tur\'an number of clique subdivisions
We show that for any integer , every properly edge-coloured graph on
vertices with more than edges contains a rainbow subdivision
of . Note that this bound on the number of edges is sharp up to the
error term. This is a rainbow analogue of some classical results on clique
subdivisions and extends some results on rainbow Tur\'an numbers. Our method
relies on the framework introduced by Sudakov and Tomon[2020] which we adapt to
find robust expanders in the coloured setting.Comment:
Towards the Erd\H{o}s-Gallai Cycle Decomposition Conjecture
In the 1960's, Erd\H{o}s and Gallai conjectured that the edges of any
-vertex graph can be decomposed into cycles and edges. We improve
upon the previous best bound of cycles and edges due to
Conlon, Fox and Sudakov, by showing an -vertex graph can always be
decomposed into cycles and edges, where is the
iterated logarithm function.Comment: Final version, accepted for publicatio