806 research outputs found
Strong Forms of Stability from Flag Algebra Calculations
Given a hereditary family of admissible graphs and a function
that linearly depends on the statistics of order-
subgraphs in a graph , we consider the extremal problem of determining
, the maximum of over all admissible
graphs of order . We call the problem perfectly -stable for a graph
if there is a constant such that every admissible graph of order
can be made into a blow-up of by changing at most
adjacencies. As special
cases, this property describes all almost extremal graphs of order within
edges and shows that every extremal graph of order is a
blow-up of .
We develop general methods for establishing stability-type results from flag
algebra computations and apply them to concrete examples. In fact, one of our
sufficient conditions for perfect stability is stated in a way that allows
automatic verification by a computer. This gives a unifying way to obtain
computer-assisted proofs of many new results.Comment: 44 pages; incorporates reviewers' suggestion
The feasible region of induced graphs
The feasible region of a graph is the collection
of points in the unit square such that there exists a sequence of
graphs whose edge densities approach and whose induced -densities
approach . A complete description of is not known
for any with at least four vertices that is not a clique or an independent
set. The feasible region provides a lot of combinatorial information about .
For example, the supremum of over all is
the inducibility of and yields the Kruskal-Katona
and clique density theorems.
We begin a systematic study of by proving some
general statements about the shape of and giving
results for some specific graphs . Many of our theorems apply to the more
general setting of quantum graphs. For example, we prove a bound for quantum
graphs that generalizes an old result of Bollob\'as for the number of cliques
in a graph with given edge density. We also consider the problems of
determining when , is a star, or is a
complete bipartite graph. In the case of our results sharpen those
predicted by the edge-statistics conjecture of Alon et. al. while also
extending a theorem of Hirst for that was proved using computer aided
techniques and flag algebras. The case of the 4-cycle seems particularly
interesting and we conjecture that is determined by
the solution to the triangle density problem, which has been solved by
Razborov.Comment: 27 page
Stability from graph symmetrisation arguments with applications to inducibility
We present a sufficient condition for the stability property of extremal
graph problems that can be solved via Zykov's symmetrisation. Our criterion is
stated in terms of an analytic limit version of the problem. We show that, for
example, it applies to the inducibility problem for an arbitrary complete
bipartite graph , which asks for the maximum number of induced copies of
in an -vertex graph, and to the inducibility problem for and
, the only complete partite graphs on at most five vertices for
which the problem was previously open.Comment: 41 page
is almost a fractalizer
We determine the maximum number of induced copies of a 5-cycle in a graph on
vertices for every . Every extremal construction is a balanced iterated
blow-up of the 5-cycle with the possible exception of the smallest level where
for , the M\"obius ladder achieves the same number of induced 5-cycles as
the blow-up of a 5-cycle on 8 vertices.
This result completes work of Balogh, Hu, Lidick\'y, and Pfender [Eur. J.
Comb. 52 (2016)] who proved an asymptotic version of the result. Similarly to
their result, we also use the flag algebra method but we extend its use to
small graphs.Comment: 24 page
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