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ON A PROBLEM OF ERDŐS ABOUT GRAPHS WHOSE SIZE IS THE TURÁN NUMBER PLUS ONE

Part of: Graph theory

Published online by Cambridge University Press:  24 May 2021

PU QIAO Open the ORCID record for PU QIAO [Opens in a new window]  and
XINGZHI ZHAN Open the ORCID record for XINGZHI ZHAN [Opens in a new window]
PU QIAO
Affiliation:
Department of Mathematics, East China University of Science and Technology, Shanghai200237, China e-mail: pq@ecust.edu.cn
XINGZHI ZHAN*
Affiliation:
Department of Mathematics, East China Normal University, Shanghai200241, China
*
e-mail: zhan@math.ecnu.edu.cn
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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.

Keywords

Turán numberextremal graph theoryErdős problembookstar4-cycle

MSC classification

Primary: 05C35: Extremal problems
Secondary: 05C30: Enumeration in graph theory 05C75: Structural characterization of families of graphs
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 105 , Issue 2 , April 2022 , pp. 177 - 187
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Copyright
© 2021 Australian Mathematical Publishing Association Inc.

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Footnotes

This research was supported by the NSFC grants 11671148 and 11771148 and Science and Technology Commission of Shanghai Municipality (STCSM) grant 18dz2271000.

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