Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-26T19:21:08.646Z Has data issue: false hasContentIssue false
  • English
  • Français

Turán Numbers of Multiple Paths and Equibipartite Forests

Published online by Cambridge University Press:  12 October 2011

NEAL BUSHAW  and
NATHAN KETTLE
NEAL BUSHAW
Affiliation:
Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA (e-mail: [email protected])
NATHAN KETTLE
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB3 0WB, UK (e-mail: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

The Turán number of a graph H, ex(n, H), is the maximum number of edges in any graph on n vertices which does not contain H as a subgraph. Let Pl denote a path on l vertices, and let k ⋅ Pl denote k vertex-disjoint copies of Pl. We determine ex(n, k ⋅ P3) for n appropriately large, answering in the positive a conjecture of Gorgol. Further, we determine ex(n, k ⋅ Pl) for arbitrary l, and n appropriately large relative to k and l. We provide some background on the famous Erdős–Sós conjecture, and conditional on its truth we determine ex(n, H) when H is an equibipartite forest, for appropriately large n.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 20 , Issue 6 , November 2011 , pp. 837 - 853
Copyright
Copyright © Cambridge University Press 2011

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]Balasubramanian, S. and Dobson, E. (2007) On the Erdős–Sós conjecture for graphs with no K 2,s. J. Graph Theory 56 301–310.CrossRefGoogle Scholar
[2]Balister, P. N., Győri, E., Lehel, J. and Schelp, R. H. (2008) Connected graphs without long paths. Discrete Math. 308 4487–4494.CrossRefGoogle Scholar
[3]Bollobás, B. (2002) Modern Graph Theory, Springer.Google Scholar
[4]Brandt, S. and Dobson, E. (1996) The Erdős–Sós conjecture for graphs of girth 5. Discrete Math. 150 411–414.CrossRefGoogle Scholar
[5]Erdős, P. and Gallai, T. (1959) On maximal paths and circuits of graphs. Acta Math. Acad. Sci. Hungar. 10 337–356.CrossRefGoogle Scholar
[6]Gorgol, I. (2011) Turán numbers for disjoint copies of graphs. Graphs Combin. 27 661–667.CrossRefGoogle Scholar
[7]Kopylov, G. N. (1977) On maximal paths and cycles in a graph. Soviet Math. Dokl. 18 593–596.Google Scholar
[8]McLennan, A. (2005) The Erdős–Sós conjecture for trees of diameter four. J. Graph Theory 49 291–301.CrossRefGoogle Scholar
[9]Moser, W. and Pach, J. (1993) Recent developments in combinatorial geometry. In New Trends in Discrete and Computational Geometry, Springer, pp. 281–302.CrossRefGoogle Scholar
[10]Saclé, J. F. and Woźniak, M. (1997) The Erdős–Sós conjecture for graphs without C 4. J. Combin. Theory Ser. B 70 367–372.CrossRefGoogle Scholar
[11]Sidorenko, A. (1989) Asymptotic solution for a new class of forbidden r-graphs. Combinatorica 9 207–215.CrossRefGoogle Scholar
[12]Simonovits, M. (1968) A method for solving extremal problems in extremal graph theory. In Theory of Graphs (Erdős, P. and Katona, G., eds), Academic Press, pp. 279–319.Google Scholar
[13]Turán, P. (1941) Egy gráfelméleti szélsőértékfeladatról. Mat. es Fiz. Lapok. 48 436–452.Google Scholar
[14]Turán, P. (1954) On the theory of graphs. Colloquium Math. 3 19–30.CrossRefGoogle Scholar
[15]Woźniak, M. (1996) On the Erdős–Sós conjecture. J. Graph Theory 21 229–234.3.0.CO;2-E>CrossRefGoogle Scholar