Modular Orientations of Random and Quasi-Random Regular Graphs

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 so 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 nontrivial eigenvalues is at most c1p and the neighborhood of each vertex contains at most c2p 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.