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Orientability Thresholds for Random Hypergraphs

Part of: Graph theory

Published online by Cambridge University Press:  23 March 2015

PU GAO  and
NICHOLAS WORMALD
PU GAO
Affiliation:
Max-Planck-Institut für Informatik, 66123 Saarbrücken, Saarland, Germany and Department of Combinatorics and Optimization, University of Waterloo, Canada (e-mail: janegao@mpi-inf.mpg.de)
NICHOLAS WORMALD
Affiliation:
Department of Combinatorics and Optimization, University of Waterloo, Canada (e-mail: nwormald@math.uwaterloo.ca)
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Abstract

Let h > w > 0 be two fixed integers. Let H be a random hypergraph whose hyperedges are all of cardinality h. To w-orient a hyperedge, we assign exactly w of its vertices positive signs with respect to the hyperedge, and the rest negative signs. A (w,k)-orientation of H consists of a w-orientation of all hyperedges of H, such that each vertex receives at most k positive signs from its incident hyperedges. When k is large enough, we determine the threshold of the existence of a (w,k)-orientation of a random hypergraph. The (w,k)-orientation of hypergraphs is strongly related to a general version of the off-line load balancing problem. The graph case, when h = 2 and w = 1, was solved recently by Cain, Sanders and Wormald and independently by Fernholz and Ramachandran. This settled a conjecture of Karp and Saks.

MSC classification

Primary: 05C80: Random graphs
Secondary: 05C85: Graph algorithms
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 24 , Issue 5 , September 2015 , pp. 774 - 824
Copyright
Copyright © Cambridge University Press 2015 

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