论文标题
二章面向图的地平线
The horizon of 2-dichromatic oriented graphs
论文作者
论文摘要
如果我们可以2色的顶点，以使每个单色部分都是无环的，则有向图的二分法数最多是2。面向图的图形是通过将其边缘定向以两个可能的方向之一将其定向的。我们研究了定向图的二分法数大于2。诺伊曼·拉拉（Neumann-Lara）发现了四个$ 7 $ vertex $ 3 $ - 二色锦标赛。我们确定$ 8 $ -VERTEX $ 3 $ - 二色锦标赛，其中不包含其中任何一个，其中有64美元。我们还找到了$ 8 $顶点上的所有$ 3 $ - diCICTIENCIENTICE的图形，其中有$ 159 $。我们确定$ 3 $ dicritioncitionpriginal的图形所具有的最小数量的弧线。有一个独特的面图，带有$ 7 $的顶点和$ 20 $弧。
The dichromatic number of a directed graph is at most 2, if we can 2-color the vertices such that each monochromatic part is acyclic. An oriented graph arises from a graph by orienting its edges in one of the two possible directions. We study oriented graphs, which have dichromatic number more than 2. Such a graph $D$ is $3$-dicritical if the removal of any arc of $D$ reduces the dichromatic number to 2. We construct infinitely many $3$-dicritical oriented graphs. Neumann-Lara found the four $7$-vertex $3$-dichromatic tournaments. We determine the $8$-vertex $3$-dichromatic tournaments, which do not contain any of these, there are $64$ of them. We also find all $3$-dicritical oriented graphs on $8$ vertices, there are $159$ of them. We determine the smallest number of arcs that a $3$-dicritical oriented graph can have. There is a unique oriented graph with $7$ vertices and $20$ arcs.
