An (r,alpha)-bounded excess flow ((r,alpha)-flow) in an orientation of a graph G=(V,E) is an assignment of a real "flow value" between 1 and r-1 to every edge. Rather than 0 as in an actual flow, some flow excess, which does not exceed alpha may accumulate in any vertex. Bounded excess flows suggest a generalization of Circular nowhere zero flows, which can be regarded as (r,0)-flows. We define (r,alpha) as Stronger or equivalent to (s,beta) If the existence of an (r,alpha)-flow in a cubic graph always implies the existence of an (s,beta)-flow in the same graph. Then we study the structure of the two-dimensional flow strength poset. A major role is played by the "Trace" parameter: tr(r,alpha)=(r-2alpha) divided by (1-alpha). Among points with the same trace the stronger is the one with the larger r (an r-cnzf is of trace r). About one half of the article is devoted to proving the main result: Every cubic graph admits a (3.5,0.5)-flow. tr(3.5,0.5)=5 so it can be considered a step in the direction of the 5-flow Conjecture. Our result is best possible for all cubic graphs while the seemingly stronger 5-flow Conjecture only applies to bridgeless graphs. We strongly rely on the notion of "k-weak bisections", defined and studied in: L. Esperet, G. Mazzuoccolo and M. Tarsi "Flows and Bisections in Cubic Graphs" J. Graph Theory, 86(2) (2017), 149-158.

中文翻译：

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三次图中的有界过剩流

图 G=(V,E) 方向上的 (r,alpha) 有界过剩流量 ((r,alpha)-flow) 是将 1 和 r-1 之间的实际“流量值”分配给每个边缘。与实际流量中的 0 不同，一些不超过 alpha 的流量过剩可能会在任何顶点累积。有界超额流量表明循环无处零流量的推广，可以将其视为 (r,0)-流量。我们将 (r,alpha) 定义为 Stronger 或等价于 (s,beta) 如果三次图中存在 (r,alpha)-flow 总是意味着同一图中存在 (s,beta)-flow . 然后我们研究了二维流动强度偏序集的结构。“Trace”参数起着主要作用：tr(r,alpha)=(r-2alpha) 除以 (1-alpha)。在具有相同轨迹的点中，具有较大 r 的点更强（r-cnzf 是轨迹 r）。大约一半的文章致力于证明主要结果：每个三次图都承认一个 (3.5,0.5) 流。tr(3.5,0.5)=5 所以它可以被认为是朝着 5-flow Conjecture 方向迈出的一步。我们的结果对于所有三次图都是最好的，而看似更强的 5-flow Conjecture 仅适用于无桥图。我们强烈依赖“k-弱二分法”的概念，在以下文献中定义和研究：L. Esperet、G. Mazzuoccolo 和 M. Tarsi “三次图中的流和二分法” J. Graph Theory, 86(2) (2017) , 149-158。我们的结果对于所有三次图都是最好的，而看似更强的 5-flow Conjecture 仅适用于无桥图。我们强烈依赖“k-弱二分法”的概念，在以下文献中定义和研究：L. Esperet、G. Mazzuoccolo 和 M. Tarsi “三次图中的流和二分法” J. Graph Theory, 86(2) (2017) , 149-158。我们的结果对所有三次图都是最好的，而看似更强的 5-flow Conjecture 仅适用于无桥图。我们强烈依赖“k-弱二分法”的概念，在以下文献中定义和研究：L. Esperet、G. Mazzuoccolo 和 M. Tarsi “三次图中的流和二分法” J. Graph Theory, 86(2) (2017) , 149-158。