Modulo k-Orientations and Tutte’s 3-Flow Conjecture in Graphs with Many Edge-Disjoint Spanning Trees

Let k be an integer, k ≥ 3, and let Zk be the cyclic group of order k. Take λk ∈ [k+2,∞) to be the smallest integer such that for every λk-edge-connected graph G and every mapping f : V (G) → Zk with |E(G)| k ≡ ∑ v∈V (G) f(v), there exits an orientation for G such that for each vertex v, d G (v) k ≡ f(v), where d G (v) denotes the out-degree of v. Lovász, Thomassen, Wu, and Zhang (2013) proved that λk ≤ 3k − 3 when k is odd and λk ≤ 3k − 2 when k is even. In this note, we prove that if a graph G contains λk − 2 edge-disjoint spanning trees, then it has an orientation such that for each vertex v, d G (v) k ≡ f(v), where f : V (G) → Zk is a mapping with |E(G)| k ≡ ∑ v∈V (G) f(v). This result confirms Tutte’s 3-Flow Conjecture in graphs with 4 edge-disjoint spanning trees.