A question at the intersection of Barnette's Hamiltonicity and Neumann-Lara's dicoloring conjecture is: Can every Eulerian oriented planar graph be vertex-partitioned into two acyclic sets? A CAI-partition of an undirected/oriented graph is a partition into a tree/connected acyclic subgraph and an independent set. Consider any plane Eulerian oriented triangulation together with its unique tripartition, i.e. partition into three independent sets. If two of these three sets induce a subgraph G that has a CAI-partition, then the above question has a positive answer. We show that if G is subcubic, then it has a CAI-partition, i.e. oriented planar bipartite subcubic 2-vertex-connected graphs admit CAI-partitions. We also show that series-parallel 2-vertex-connected graphs admit CAI-partitions. Finally, we present a Eulerian oriented triangulation such that no two sets of its tripartition induce a graph with a CAI-partition. This generalizes a result of Alt, Payne, Schmidt, and Wood to the oriented setting.

翻译：巴尼特哈密顿性与纽曼-拉拉有向着色猜想交叉领域的一个问题是：每个欧拉有向平面图是否都能被顶点分割为两个无环集？无向/有向图的CAI分割是指将其分割为一棵树/连通无环子图与一个独立集。考虑任意平面欧拉有向三角剖分及其唯一的三分割（即分割为三个独立集）。若这三个集合中有两个诱导出的子图G具有CAI分割，则上述问题存在肯定答案。我们证明：若G是次三次的，则其具有CAI分割，即有向平面二分次三次2-顶点连通图允许CAI分割。我们还证明：串并联2-顶点连通图允许CAI分割。最后，我们构造了一个欧拉有向三角剖分，其三分割中任意两个集合诱导出的子图均不具有CAI分割。这将阿尔特、佩恩、施密特和伍德的结果推广至有向图情形。