Toward Grünbaum's Conjecture

Abstract

Given a spanning tree $T$ of a planar graph $G$, the co-tree of $T$ is the spanning tree of the dual graph $G^{\ast }$ with edge set $(E(G)-E(T){)}^{\ast }$. Grünbaum 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 Grünbaum'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.