188 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
On the inducibility of small trees
The quantity that captures the asymptotic value of the maximum number of
appearances of a given topological tree (a rooted tree with no vertices of
outdegree ) with leaves in an arbitrary tree with sufficiently large
number of leaves is called the inducibility of . Its precise value is known
only for some specific families of trees, most of them exhibiting a symmetrical
configuration. In an attempt to answer a recent question posed by Czabarka,
Sz\'ekely, and the second author of this article, we provide bounds for the
inducibility of the -leaf binary tree whose branches are a
single leaf and the complete binary tree of height . It was indicated before
that appears to be `close' to . We can make this precise by
showing that . Furthermore, we
also consider the problem of determining the inducibility of the tree ,
which is the only tree among -leaf topological trees for which the
inducibility is unknown
On the inducibility of cycles
In 1975 Pippenger and Golumbic proved that any graph on vertices admits
at most induced -cycles. This bound is larger by a
multiplicative factor of than the simple lower bound obtained by a blow-up
construction. Pippenger and Golumbic conjectured that the latter lower bound is
essentially tight. In the present paper we establish a better upper bound of
. This constitutes the first progress towards proving
the aforementioned conjecture since it was posed
Maximising the number of induced cycles in a graph
We determine the maximum number of induced cycles that can be contained in a
graph on vertices, and show that there is a unique graph that
achieves this maximum. This answers a question of Tuza. We also determine the
maximum number of odd or even cycles that can be contained in a graph on vertices and characterise the extremal graphs. This resolves a conjecture
of Chv\'atal and Tuza from 1988.Comment: 36 page
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