In this paper we study the flow properties of graphs containing a spanning triangle-tree. Our main results provide a structure characterization of graphs with a spanning triangle-tree admitting a nowhere-zero 3-flow. All these graphs without nowhere-zero 3-flows are constructed from \(K_4\) by a so-called bull-growth operation. This generalizes a result of Fan et al. in 2008 on triangularly-connected graphs and particularly shows that every 4-edge-connected graph with a spanning triangle-tree has a nowhere-zero 3-flow. A well-known classical theorem of Jaeger in 1979 shows that every graph with two edge-disjoint spanning trees admits a nowhere-zero 4-flow. We prove that every graph with two edge-disjoint spanning triangle-trees has a flow strictly less than 3.

在本文中，我们研究了包含生成三角树的图的流动特性。我们的主要结果提供了具有跨度三角树的图的结构表征，该三角树允许零位零流。所有这些无处无零的3个流的图都是通过所谓的牛生长操作从\（K_4 \）构造的。这概括了范等人的结果。在2008年的三角形连接图上，它特别显示了每个带有跨接三角树的4边连接图都有无处零的3流。1979年Jaeger的一个著名的经典定理表明，每个带有两个不相交的生成树的图都接受无处零的4流。我们证明每个带有两个不相交的生成三角树的图的流量严格小于3。