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The Proofs of Two Directed Paths Conjectures of Bollobás and Leader

Part of: Extremal combinatorics Graph theory

Published online by Cambridge University Press:  22 May 2017

TREVOR PINTO
TREVOR PINTO*
Affiliation:
School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, UK
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Abstract

Let A and B be disjoint sets, of size 2k, of vertices of Qn, the n-dimensional hypercube. In 1997, Bollobás and Leader proved that there must be (n − k)2k edge-disjoint paths between such A and B. They conjectured that when A is a down-set and B is an up-set, these paths may be chosen to be directed (that is, the vertices in the path form a chain). We use a novel type of compression argument to prove stronger versions of these conjectures, namely that the largest number of edge-disjoint paths between a down-set A and an up-set B is the same as the largest number of directed edge-disjoint paths between A and B. Bollobás and Leader made an analogous conjecture for vertex-disjoint paths, and we prove a strengthening of this by similar methods. We also prove similar results for all other sizes of A and B.

MSC classification

Primary: 05C38: Paths and cycles
Secondary: 05D05: Extremal set theory
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 26 , Issue 5 , September 2017 , pp. 762 - 774
Copyright
Copyright © Cambridge University Press 2017 

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