 
Summary: Modular orientations of random and quasirandom regular graphs
Noga Alon
Pawel Pralat
Abstract
Extending an old conjecture of Tutte, Jaeger conjectured in 1988 that for any fixed integer
p 1, the edges of any 4pedge 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, 4pedge 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 quasirandom properties, namely, the
absolute value of each of their nontrivial eigenvalues is at most c1p2/3
and the neighborhood
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.
1 Introduction
A nowherezero 3flow in an undirected graph G = (V, E) is an orientation of its edges and a
function f assigning a number f(e) {1, 2} to any oriented edge e such that for any vertex
v V ,
