论文标题
两个强的$ 3 $流量定理用于平面图
Two Strong $3$-Flow Theorems for Planar Graphs
论文作者
论文摘要
1972年，图特（Tutte）提出了$ 3 $ - 流的猜想：所有$ 4 $ - 边缘连接的图形都没有零$ 3 $ - 流。 Jaeger等人（1992）将其扩展为允许顶点在流入和流出之间具有规定的，可能是非零的差异（Modulo $ 3 $）。他们推测所有具有有效处方功能的$ 5 $ - 边缘连接的图形都没有零$ 3 $ - 流的处方会议（我们将其称为强$ 3 $ - 流的猜想）。 Kochol（2001）表明，用$ 5 $ - 边缘连接的$ 4 $ edge连接足以证明$ 3 $ - 流量猜想和Lovász等人（2013年）表明，如果Egge Connective to Exge Connective to to Egge Connective to to Exge Connective to to Exge Connective to hamlees to to to the Egge connective to phow to the Exge connective to hloseed to to the Exge connective to 6 $ $ 6 $ - $ 6 $ - $ 6 $ - $ 6 $。这两个问题仍然以$ 5 $ - 边缘连接的图形开放。 $ 3 $流量的猜想是为平面图而闻名的，因为它是Grötzsch的颜色定理的双重。 Steinberg and Younger（1989）使用平面图的流量提供了第一个直接证明，以及用于投影平面图的证明。 Richter等人（2016年）使用强大的$ 3 $流量猜想的平面图提供了第一个直接证明。我们为它们的结果提供了两个扩展，我们开发了它们，以证明投影平面图的强劲$ 3 $流量猜想。
In 1972, Tutte posed the $3$-Flow Conjecture: that all $4$-edge-connected graphs have a nowhere zero $3$-flow. This was extended by Jaeger et al.(1992) to allow vertices to have a prescribed, possibly non-zero difference (modulo $3$) between the inflow and outflow. They conjectured that all $5$-edge-connected graphs with a valid prescription function have a nowhere zero $3$-flow meeting that prescription (we call this the Strong $3$-Flow Conjecture). Kochol (2001) showed that replacing $4$-edge-connected with $5$-edge-connected would suffice to prove the $3$-Flow Conjecture and Lovász et al.(2013) showed that the $3$-Flow and Strong $3$-Flow Conjectures hold if the edge connectivity condition is relaxed to $6$-edge-connected. Both problems are still open for $5$-edge-connected graphs. The $3$-Flow Conjecture was known to hold for planar graphs, as it is the dual of Grötzsch's Colouring Theorem. Steinberg and Younger (1989) provided the first direct proof using flows for planar graphs, as well as a proof for projective planar graphs. Richter et al.(2016) provided the first direct proof using flows of the Strong $3$-Flow Conjecture for planar graphs. We provide two extensions to their result, that we developed in order to prove the Strong $3$-Flow Conjecture for projective planar graphs.
