Abstract
Abstract
We consider finite simple graphs. Given a graph H and a positive integer
$n,$
the Turán number of H for the order
$n,$
denoted
$\mathrm {ex}(n,H),$
is the maximum size of a graph of order n not containing H as a subgraph. Erdős asked: ‘For which graphs H is it true that every graph on n vertices and
$\mathrm {ex}(n,H)+1$
edges contains at least two H’s? Perhaps this is always true.’ We solve this problem in the negative by proving that for every integer
$k\ge 4$
there exists a graph H of order k and at least two orders n such that there exists a graph of order n and size
$\mathrm {ex}(n,H)+1$
which contains exactly one copy of
$H.$
Denote by
$C_4$
the
$4$
-cycle. We also prove that for every integer n with
$6\le n\le 11$
there exists a graph of order n and size
$\mathrm {ex}(n,C_4)+1$
which contains exactly one copy of
$C_4,$
but, for
$n=12$
or
$n=13,$
the minimum number of copies of
$C_4$
in a graph of order n and size
$\mathrm {ex}(n,C_4)+1$
is two.
Publisher
Cambridge University Press (CUP)
Reference13 articles.
1. Some theorems on graphs;Erdős;Riveon Lematematika,1955
2. Graphs without four-cycles
3. Extremal graphs without 4-cycles
4. Problem 28;Mantel;Wisk. Opgaven,1907
Cited by
3 articles.
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