Modular orientations of random and quasi-random regular graphs

  • Authors:

  • Noga Alon;PaweŁ PraŁat

  • Affiliations:

  • 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: nogaa@tau.ac ...;Department of mathematics, west virginia university, morgantown, wv 26506-6310, usa (e-mail: pralat@math.wvu.edu)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2011

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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.