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On triangle-free graphs maximizing embeddings of bipartite graphs
Authors
Dmitriy Gorovoy
Andrzej Grzesik
Justyna Jaworska
Publication date
22 September 2023
Publisher
View
on
arXiv
Abstract
In 1991 Gy\H ori, Pach, and Simonovits proved that for any bipartite graph
H
H
H
containing a matching avoiding at most 1 vertex, the maximum number of copies of
H
H
H
in any large enough triangle-free graph is achieved in a balanced complete bipartite graph. In this paper we improve their result by showing that if
H
H
H
is a bipartite graph containing a matching of size
x
x
x
and at most
1
2
x
β
1
\frac{1}{2}\sqrt{x-1}
2
1
β
x
β
1
β
unmatched vertices, then the maximum number of copies of
H
H
H
in any large enough triangle-free graph is achieved in a complete bipartite graph. We also prove that such a statement cannot hold if the number of unmatched vertices is
Ξ©
(
x
)
\Omega(x)
Ξ©
(
x
)
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oai:arXiv.org:2309.12866
Last time updated on 12/10/2023