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Modular Orientations of Random and Quasi-Random Regular Graphs

Published online by Cambridge University Press:  27 January 2011

NOGA ALON  and
PAWEŁ PRAŁAT
NOGA ALON
Affiliation:
Sackler School of Mathematics and Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel and Institute for Advanced Study, Princeton, NJ 08540, USA (e-mail: [email protected])
PAWEŁ PRAŁAT
Affiliation:
Department of Mathematics, West Virginia University, Morgantown, WV 26506-6310, USA (e-mail: [email protected])
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Abstract

Extending an old conjecture of Tutte, Jaeger conjectured in 1988 that for any fixed integer p ≥ 1, the edges of any 4p-edge connected graph can be oriented so that the difference between the outdegree and the indegree of each vertex is divisible by 2p+1. It is known that it suffices to prove this conjecture for (4p+1)-regular, 4p-edge connected graphs. Here we show that there exists a finite p0 such that for every p > p0 the assertion of the conjecture holds for all (4p+1)-regular graphs that satisfy some mild quasi-random properties, namely, the absolute value of each of their non-trivial eigenvalues is at most c1p2/3 and the neighbourhood of each vertex contains at most c2p3/2 edges, where c1, c2 > 0 are two absolute constants. In particular, this implies that for p > p0 the assertion of the conjecture holds asymptotically almost surely for random (4p+1)-regular graphs.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 20 , Issue 3 , May 2011 , pp. 321 - 329
Copyright
Copyright © Cambridge University Press 2011

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