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This paper considers the conjecture by Gr\"unbaum that every planar3-connected graph has a spanning tree $T$ such that both $T$ and its co-treehave maximum degree at most 3. Here, the co-tree of $T$ is the spanning tree ofthe dual obtained by taking the duals of the non-tree edges. While Gr\"unbaum'sconjecture remains open, we show that every planar 3-connected graph has aspanning tree $T$ such that both $T$ and its co-tree have maximum degree atmost 5. It can be found in linear time.

Trees and co-trees with constant maximum degree in planar 3-connected graphs