Year: 2026
Type: preprint
Abstract: Nash--Williams's orientation theorem converts undirected edge-connectivity into directed connectivity. We extend it to prescribed outdegrees modulo $q$, whose residues must sum to the number of edges modulo $q$. For integers $q,k\ge1$, every such prescription has a realization with at least $k$ edges in each direction across each nontrivial cut if the graph is $(2q+2k-2)$-edge-connected or contains $q+2k-1$ edge-disjoint spanning trees. The method combines a cycle-space approximation bound with rounding that preserves the imbalance across every cut. Fractional spanning-tree packing gives an exact criterion: every compatible prescription has a strongly connected realization if and only if the fractional packing number exceeds $q$. This yields a polynomial-time recognition test. With no directed connectivity requirement, $q$ disjoint spanning trees suffice, proving Hasanvand's prescribed-residue conjecture. For every odd prime $p$, this proves the conjectured statement that the union of $p$ graphic bases over the field with $p$ elements is an additive basis. For every integer $p\ge5$, the theorem and the counterexamples of Han, Li, Wu and Zhang give the exact edge-connectivity threshold $4p+2$ for a modulo-$(2p+1)$ orientation. We also show that every six-edge-connected graph has circular flow number strictly below three, as conjectured by Li, Thomassen, Wu and Zhang. For three-flow-critical graphs on $n$ vertices and $m$ edges, we improve the previous general bound $m\le4n-10$ to $m\le3n-6$, and to $m\le3n-7$ when $n\ge5$. Further applications include Seymour's six-flow theorem, star decompositions, antisymmetric flows, and signed modular orientations.
Source: arXiv (Cornell University)
Author: Jafari, Amir
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7220919875