Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-26T19:48:40.578Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
XINGZHI ZHAN*
Affiliation:
Department of Mathematics, East China Normal University, Shanghai200241, China
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© 2021 Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

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

References

Bollobás, B., Extremal Graph Theory (Academic Press, London–New York, 1978).Google Scholar
Chartrand, G., Lesniak, L. and Zhang, P., Graphs and Digraphs, 6th edn (CRC Press, Boca Raton, FL, 2016).Google Scholar
Clapham, C. R. J., Flockhart, A. and Sheehan, J., ‘Graphs without four-cycles’, J. Graph Theory 13(1) (1989), 29–47.10.1002/jgt.3190130107CrossRefGoogle Scholar
Erdős, P., ‘Some theorems on graphs’, Riveon Lematematika 9 (1955), 13–17.Google Scholar
Erdős, P., ‘Some of my favourite unsolved problems’, in: A Tribute to Paul Erdős (eds. Baker, A., Bollabás, B. and Hajnal, A.) (Cambridge University Press, Cambridge, 1990).Google Scholar
Firke, F. A., Kosek, P. M., Nash, E. D. and Williford, J., ‘Extremal graphs without 4-cycles’, J. Combin. Theory Ser. B 103(3) (2013), 327–336.10.1016/j.jctb.2013.01.001CrossRefGoogle Scholar
Füredi, Z., ‘Graphs without quadrilaterals’, J. Combin. Theory Ser. B 34(2) (1983), 187–190.10.1016/0095-8956(83)90018-7CrossRefGoogle Scholar
Füredi, Z., Naor, A. and Verstraëte, J., ‘On the Turán number for the hexagon’, Adv. Math. 203(6) (2006), 476–496.10.1016/j.aim.2005.04.011CrossRefGoogle Scholar
He, J., Ma, J. and Yang, T., ‘Stability and supersaturation of 4-cycles’, Preprint, 2021, arXiv:1912.00986v3.Google Scholar
Mantel, W., ‘Problem 28’, Wisk. Opgaven 10 (1907), 60–61.Google Scholar
Moon, J. W., ‘On the number of complete subgraphs of a graph’, Canad. Math. Bull. 8 (1965), 831–834.10.4153/CMB-1965-065-9CrossRefGoogle Scholar
Turán, P., ‘Eine Extremalaufgabe aus der Graphentheorie’, Mat. Fiz. Lapok 48 (1941), 436–452.Google Scholar
Yan, J. and Zhan, X., ‘The Turán number of book graphs’, Preprint, 2021, arXiv:2010.09973v1.Google Scholar