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On Subtrees of Directed Graphs with No Path of Length Exceeding One

Published online by Cambridge University Press:  20 November 2018

R. L. Graham
R. L. Graham*
Affiliation:
Bell Telephone Laboratories, Inc., Murray Hill, New Jersey
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The following theorem was conjectured to hold by P. Erdös [1]:

Theorem 1. For each finite directed tree T with no directed path of length 2, there exists a constant c(T) such that if G is any directed graph with n vertices and at least c(T)n edges and n is sufficiently large, then T is a subgraph of G.

In this note we give a proof of this conjecture. In order to prove Theorem 1, we first need to establish the following weaker result.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 13 , Issue 3 , September 1970 , pp. 329 - 332
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Erdös, P., (personal communication).Google Scholar