154 research outputs found
Rectilinear approximation and volume estimates for hereditary bodies via [0,1]-decorated containers
We use the hypergraph container theory of Balogh--Morris--Samotij and
Saxton--Thomason to obtain general rectilinear approximations and volume
estimates for sequences of bodies closed under certain families of projections.
We give a number of applications of our results, including a multicolour
generalisation of a theorem of Hatami, Janson and Szegedy on the entropy of
graph limits. Finally, we raise a number of questions on geometric and analytic
approaches to containers.Comment: 25 pages, author accepted manuscript, to appear in Journal of Graph
Theor
Supersaturation for hereditary properties
Let be a collection of -uniform hypergraphs, and let . It is known that there exists such that the
probability of a random -graph in not containing an induced
subgraph from is . Let each graph in
have at least vertices. We show that in fact for every
, there exists
such that the probability of a random -graph in containing less
than induced subgraphs each lying in is at most
.
This statement is an analogue for hereditary properties of the
supersaturation theorem of Erd\H{o}s and Simonovits. In our applications we
answer a question of Bollob\'as and Nikiforov.Comment: 5 pages, submitted to European Journal of Combinatoric
The t-stability number of a random graph
Given a graph G = (V,E), a vertex subset S is called t-stable (or
t-dependent) if the subgraph G[S] induced on S has maximum degree at most t.
The t-stability number of G is the maximum order of a t-stable set in G. We
investigate the typical values that this parameter takes on a random graph on n
vertices and edge probability equal to p. For any fixed 0 < p < 1 and fixed
non-negative integer t, we show that, with probability tending to 1 as n grows,
the t-stability number takes on at most two values which we identify as
functions of t, p and n. The main tool we use is an asymptotic expression for
the expected number of t-stable sets of order k. We derive this expression by
performing a precise count of the number of graphs on k vertices that have
maximum degree at most k. Using the above results, we also obtain asymptotic
bounds on the t-improper chromatic number of a random graph (this is the
generalisation of the chromatic number, where we partition of the vertex set of
the graph into t-stable sets).Comment: 25 pages; v2 has 30 pages and is identical to the journal version
apart from formatting and a minor amendment to Lemma 8 (and its proof on p.
21
Largest sparse subgraphs of random graphs
For the Erd\H{o}s-R\'enyi random graph G(n,p), we give a precise asymptotic
formula for the size of a largest vertex subset in G(n,p) that induces a
subgraph with average degree at most t, provided that p = p(n) is not too small
and t = t(n) is not too large. In the case of fixed t and p, we find that this
value is asymptotically almost surely concentrated on at most two explicitly
given points. This generalises a result on the independence number of random
graphs. For both the upper and lower bounds, we rely on large deviations
inequalities for the binomial distribution.Comment: 15 page
- …