AI 中文总结
本文通过生成树打包研究 Tutte 流猜想，提出指定方向定理并证明图论加法基猜想，同时给出圆周流界及反例说明打包数大于 9/2 不足以保证模五方向。
AI 中文摘要
Tutte 的流猜想询问一个图何时具有小整数值的无处为零的流。我们通过生成树打包来处理这些问题。一个常见的分析论证恢复了 Seymour 的六流定理、六边连通图的三流定理以及具有两条边不相交生成树的图的四流定理。其主要新结论是一个指定方向定理：对于每个整数 $q\ge2$，$q$ 条边不相交的生成树足以实现模 $q$ 的每个兼容的出度指定。对于奇素数模数，这证明了图论加法基猜想。一个具有指定边值范围的版本给出了来自树打包的尖锐圆周流界，以及一个以圈秩表示的界，该界对于连通的无桥图严格低于六。我们还考察了五流猜想在将三边连通三次图的每条边三倍化所得到的图上的一个重新表述。一个具有分数生成树打包数 $41/9$ 且没有模五方向的 $46$ 顶点图表明，仅打包数大于 $9/2$ 并不能保证任意图具有这样的方向。证明结合了 Alon、Bucić 和 Davies 的极小化方法与积分舍入。
英文摘要
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, Bucić, 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.
