16 research outputs found
On the number of 4-cycles in a tournament
If is an -vertex tournament with a given number of -cycles, what
can be said about the number of its -cycles? The most interesting range of
this problem is where is assumed to have cyclic triples for
some and we seek to minimize the number of -cycles. We conjecture that
the (asymptotic) minimizing is a random blow-up of a constant-sized
transitive tournament. Using the method of flag algebras, we derive a lower
bound that almost matches the conjectured value. We are able to answer the
easier problem of maximizing the number of -cycles. These questions can be
equivalently stated in terms of transitive subtournaments. Namely, given the
number of transitive triples in , how many transitive quadruples can it
have? As far as we know, this is the first study of inducibility in
tournaments.Comment: 11 pages, 5 figure
Graphs with few 3-cliques and 3-anticliques are 3-universal
For given integers k, l we ask whether every large graph with a sufficiently
small number of k-cliques and k-anticliques must contain an induced copy of
every l-vertex graph. Here we prove this claim for k=l=3 with a sharp bound. A
similar phenomenon is established as well for tournaments with k=l=4.Comment: 12 pages, 1 figur
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
The exact minimum number of triangles in graphs of given order and size
What is the minimum number of triangles in a graph of given order and size?
Motivated by earlier results of Mantel and Tur\'an, Rademacher solved the first
non-trivial case of this problem in 1941. The problem was revived by Erd\H{o}s
in 1955; it is now known as the Erd\H{o}s-Rademacher problem. After attracting
much attention, it was solved asymptotically in a major breakthrough by
Razborov in 2008. In this paper, we provide an exact solution for all large
graphs whose edge density is bounded away from~, which in this range
confirms a conjecture of Lov\'asz and Simonovits from 1975. Furthermore, we
give a description of the extremal graphs.Comment: Published in Forum of Mathematics, Pi, Volume 8, e8 (2020