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 $G^*$ with edge set $(E(G)-E(T))^*$. 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.

Figures (7)

• Figure 1: Properties of Schnyder woods. Condition \ref{['def:Schnyderwood']}\ref{['def:Schnyderwood3']} at a vertex.
• Figure 2: The completion of $G$ obtained by superimposing $G^\sigma$ and its suspended dual $G^{\sigma^*}$ (the latter depicted with dotted edges). The primal Schnyder wood is not the minimal element of the lattice of Schnyder woods of $G$, as this completion contains a clockwise directed cycle (marked in yellow).
• Figure 3: The clockwise cycle of $\widetilde{G}_S$ of the proof of Lemma \ref{['lem_all_edges_right']}, depicted in yellow.
• Figure 4: Illustration for some of the definitions used in Theorem \ref{['thm_max_deg_4']}. If Case 1 applies to $P_c$, we add the edges marked in yellow to $D$.
• Figure 5: If $v_kv_{k+1}$ is 2-colored, then $\widetilde{G}_S$ contains a clockwise cycle (depicted in yellow).

Theorems & Definitions (22)

• Conjecture : Grünbaum Grunbaum1970, 1970
• Definition 1
• Lemma 2: Schnyder1990Felsner2004a
• Lemma 3: Felsner Felsner2001
• Definition 4
• Lemma 5: Kant1992Felsner2004
• Corollary 6
• Lemma 7: Mendez1994Felsner2004aFusy2007
• Definition 8
• Lemma 9