Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-28T16:12:58.599Z Has data issue: false hasContentIssue false
  • English
  • Français

Extremal Numbers for Odd Cycles

Published online by Cambridge University Press:  01 December 2014

ZOLTAN FÜREDI  and
DAVID S. GUNDERSON
ZOLTAN FÜREDI
Affiliation:
Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda utca 13–15, H-1053, Budapest, Hungary (e-mail: z-furedi@illinois.edu, furedi.zoltan@renyi.mta.hu)
DAVID S. GUNDERSON
Affiliation:
Department of Mathematics, University of Manitoba, 521 Machray Hall, Winnipeg, Manitoba R3T 2N2, Canada (e-mail: David.Gunderson@umanitoba.ca)
Get access Rights & Permissions [Opens in a new window]

Abstract

We describe the C2k+1-free graphs on n vertices with maximum number of edges. The extremal graphs are unique for n ∉ {3k − 1, 3k, 4k − 2, 4k − 1}. The value of ex(n, C2k+1) can be read out from the works of Bondy [3], Woodall [14], and Bollobás [1], but here we give a new streamlined proof. The complete determination of the extremal graphs is also new.

We obtain that the bound for n0(C2k+1) is 4k in the classical theorem of Simonovits, from which the unique extremal graph is the bipartite Turán graph.

Keywords

Primary 05C35Secondary 05D99
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 24 , Special Issue 4: Oberwolfach Special Issue Part 1 , July 2015 , pp. 641 - 645
Copyright
Copyright © Cambridge University Press 2014 

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.)

References

[1]Bollobás, B. (1979) Extremal Graph Theory, Academic Press.CrossRefGoogle Scholar
[2]Bondy, J. A. (1971) Pancyclic graphs I. J. Combin. Theory Ser. B 11 80–84.CrossRefGoogle Scholar
[3]Bondy, J. A. (1971) Large cycles in graphs. Discrete Math. 1 121–132.CrossRefGoogle Scholar
[4]Brandt, S. (1997) A sufficient condition for all short cycles. In 4th Twente Workshop on Graphs and Combinatorial Optimization. Discrete Appl. Math. 79 63–66.CrossRefGoogle Scholar
[5]Dzido, T. (2013) A note on Turán numbers for even wheels. Graphs Combin. 29 1305–1309.CrossRefGoogle Scholar
[6]Erdős, P. and Gallai, T. (1959) On maximal paths and circuits of graphs. Acta Math. Acad. Sci. Hungar. 10 337–356.CrossRefGoogle Scholar
[7]Faudree, R. J. and Schelp, R. H. (1975) Path Ramsey numbers in multicolorings. J. Combin. Theory Ser. B 19 150–160.CrossRefGoogle Scholar
[8]Füredi, Z. and Simonovits, M. (2013) The history of degenerate (bipartite) extremal graph problems. In Erdős Centennial (Lovász, L.et al., eds), Vol. 25 of Bolyai Society Mathematical Studies, pp. 167–262.CrossRefGoogle Scholar
[9]Kopylov, G. N. (1977) Maximal paths and cycles in a graph. Dokl. Akad. Nauk SSSR 234 19–21. (English translation in Soviet Math. Dokl. 18 (1977) 593–596.)Google Scholar
[10]Mantel, W. (1907) Solution to Problem 28, by H. Gouwentak, W. Mantel, J. Teixeira de Mattes, F. Schuh, and W. A. Wythoff. Wiskundige Opgaven 10 60–61.Google Scholar
[11]Simonovits, M. (1974) Extremal graph problems with symmetrical extremal graphs: Additional chromatic conditions. Discrete Math. 7 349–376.CrossRefGoogle Scholar
[12]Turán, P. (1941) Eine Extremalaufgabe aus der Graphentheorie (in Hungarian). Math. Fiz Lapok 48 436–452.Google Scholar
[13]Turán, P. (1954) On the theory of graphs. Colloq. Math. 3 19–30.CrossRefGoogle Scholar
[14]Woodall, D. R. (1972) Sufficient conditions for circuits in graphs. Proc. London Math. Soc. (3) 24 739–755.CrossRefGoogle Scholar
[15]Woodall, D. R. (1976) Maximal circuits of graphs I. Acta Math. Acad. Sci. Hungar. 28 77–80.CrossRefGoogle Scholar