Grünbaum’s conjecture on degree-3 spanning tree and co-tree in planar 3-connected graphs

Establish whether every planar 3-connected graph G contains a spanning tree T such that both T and the co-tree in the planar dual G*—defined as the spanning tree with edge set (E(G) − E(T))*—have maximum degree at most 3.

Background

Barnette’s theorem guarantees a 3-tree (a spanning tree of maximum degree at most 3) in every planar 3-connected graph. By cut–cycle duality, the complement of any spanning tree in a planar graph corresponds to a spanning tree in its planar dual, termed the co-tree. Grünbaum proposed combining these two structures by requiring both the tree and co-tree to have degree at most 3.

Partial progress is known: Biedl proved the existence of a spanning tree whose tree and co-tree both have maximum degree at most 5, and this paper improves the bound to 4 using structural properties of Schnyder woods. The original degree-3 assertion, however, is not settled and remains a central open problem in this area.