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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: nobushaw@memphis.edu)
NATHAN KETTLE
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB3 0WB, UK (e-mail: n.kettle@dpmms.cam.ac.uk)
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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

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