1,763 research outputs found
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
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
Inducibility of directed paths
A long standing open problem in extremal graph theory is to describe all graphs that maximize the number of induced copies of a path on four vertices. The character of the problem changes in the setting of oriented graphs, and becomes more tractable. Here we resolve this problem in the setting of oriented graphs without transitive triangles
On the maximum number of odd cycles in graphs without smaller odd cycles
We prove that for each odd integer , every graph on vertices
without odd cycles of length less than contains at most cycles of
length . This generalizes the previous results on the maximum number of
pentagons in triangle-free graphs, conjectured by Erd\H{o}s in 1984, and
asymptotically determines the generalized Tur\'an number
for odd . In contrary to the previous results
on the pentagon case, our proof is not computer-assisted
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