Tree packing and the graphic Additive Base Conjecture

Abstract

Tutte's flow conjectures ask when a graph has a nowhere-zero flow with small integer values. We approach these questions through spanning-tree packing and an analytic minimization argument of Alon, Buci\'c, and Davies. For every integer $q\ge2$, we prove that $q$ edge-disjoint spanning trees suffice to realize every compatible outdegree prescription modulo $q$. For odd prime moduli this proves the graphic Additive Base Conjecture. More generally, integer intervals of edge values realize every prescribed boundary when their widths satisfy the corresponding tree-packing inequalities. We strengthen this condition by allowing a deficit of two in every partition inequality, provided every cut has total width at least $q-1$. The allowance of two is sharp for $q\ge3$. Consequently, five-edge-connected graphs with at most ten odd-degree vertices are $\mathbb Z_3$-connected, and three-edge-connected graphs with at most fourteen odd-degree vertices are $\mathbb Z_5$-connected. Every orientation of a nine-edge-connected graph with at most fourteen odd-degree vertices admits an antisymmetric $\mathbb Z_5$-flow. The refinement also gives sharp bounds for extending preorientations after deleting edges or vertices, including the boundary case of $2q$-edge-connectivity. The same argument recovers Seymour's six-flow theorem, the three-flow theorem for six-edge-connected graphs, and the four-flow theorem for graphs with two edge-disjoint spanning trees. It also gives sharp circular-flow bounds and a cycle-rank bound strictly below six. We examine a five-flow reformulation obtained by tripling the edges of cubic graphs. A $46$-vertex graph of fractional packing number $41/9$ with no modulo-five orientation shows that packing greater than $9/2$ alone does not guarantee such an orientation in arbitrary graphs.