Given a spanning tree $T$ of a planar graph $G$, the co-tree of $T$ is the spanning tree of the dual graph $G^*$ with edge set $(E(G)-E(T))^*$. Gr\"unbaum conjectured in 1970 that every planar 3-connected graph $G$ contains a spanning tree $T$ such that both $T$ and its co-tree have maximum degree at most 3. While Gr\"unbaum's conjecture remains open, Biedl proved that there is a spanning tree $T$ such that $T$ and its co-tree have maximum degree at most 5. By using new structural insights into Schnyder woods, we prove that there is a spanning tree $T$ such that $T$ and its co-tree have maximum degree at most 4.

翻译：给定平面图$G$的一棵生成树$T$，$T$的余树是对偶图$G^*$中以边集$(E(G)-E(T))^*$构成的生成树。Grünbaum于1970年猜想：每个平面3-连通图$G$都存在一棵生成树$T$，使得$T$及其余树的最大度数均不超过3。尽管Grünbaum猜想仍未解决，Biedl已证明存在生成树$T$使得$T$及其余树的最大度数不超过5。通过利用Schnyder woods的新结构洞见，我们证明存在生成树$T$使得$T$及其余树的最大度数不超过4。