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Trees and co-trees in planar 3-connected graphs An easier proof via Schnyder woods

Christian Ortlieb, Jens M. Schmidt

TL;DR

The paper addresses Grünbaum's conjecture, which asks for a spanning tree in every $3$-connected planar graph whose co-tree also has maximum degree $3$. It offers an easier, Schnyder-wood–driven proof of a related result due to Biedl, showing that a spanning tree can be chosen so that both it and its co-tree have maximum degree at most $5$. The approach builds a graph $H(G)$ from a Schnyder wood on the suspension $G^\sigma$ and its dual $H^\circ(G^*)$, proves degree bounds and connectivity, and then applies a minimum-weight spanning-tree argument in the style of Biedl to obtain the desired tree and co-tree properties. The method uses dual Schnyder woods and ordered path partitions to relate primal and dual structures, offering a streamlined framework that could shine light on Grünbaum's conjecture.

Abstract

Let $G$ be a 3-connected planar graph. Define the co-tree of a spanning tree $T$ of $G$ as the graph induced by the dual edges of $E(G)-E(T)$. The well-known cut-cycle duality implies that the co-tree is itself a tree. Let a $k$-tree be a spanning tree with maximum degree $k$. In 1970, Grünbaum conjectured that every 3-connected planar graph contains a 3-tree whose co-tree is also a 3-tree. In 2014, Biedl showed that every such graph contains a 5-tree whose co-tree is a 5-tree. In this paper, we present an easier proof of Biedl's result

Trees and co-trees in planar 3-connected graphs An easier proof via Schnyder woods

TL;DR

The paper addresses Grünbaum's conjecture, which asks for a spanning tree in every -connected planar graph whose co-tree also has maximum degree . It offers an easier, Schnyder-wood–driven proof of a related result due to Biedl, showing that a spanning tree can be chosen so that both it and its co-tree have maximum degree at most . The approach builds a graph from a Schnyder wood on the suspension and its dual , proves degree bounds and connectivity, and then applies a minimum-weight spanning-tree argument in the style of Biedl to obtain the desired tree and co-tree properties. The method uses dual Schnyder woods and ordered path partitions to relate primal and dual structures, offering a streamlined framework that could shine light on Grünbaum's conjecture.

Abstract

Let be a 3-connected planar graph. Define the co-tree of a spanning tree of as the graph induced by the dual edges of . The well-known cut-cycle duality implies that the co-tree is itself a tree. Let a -tree be a spanning tree with maximum degree . In 1970, Grünbaum conjectured that every 3-connected planar graph contains a 3-tree whose co-tree is also a 3-tree. In 2014, Biedl showed that every such graph contains a 5-tree whose co-tree is a 5-tree. In this paper, we present an easier proof of Biedl's result
Paper Structure (6 sections, 7 theorems, 5 figures)

This paper contains 6 sections, 7 theorems, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dual Schnyder Woods.
  4. Ordered Path Partitions.
  5. Our alternative proof
  6. Conclusion

Key Result

Lemma 3

For every Schnyder wood $S$ of $G^\sigma$, $S^*$ is a Schnyder wood of $G^{\sigma^*}$.

Figures (5)

  • Figure 1: Example for Condition \ref{['def:Schnyderwood']}\ref{['def:Schnyderwood3']} at a vertex in a Schnyder wood. The incoming edges in color $i$ are in the clockwise sector between the outgoing edge in color $i+1$ and the outgoing edge in color $i-1$.
  • Figure 2: A Schnyder wood of the suspension of a 3-connected planar graph.
  • Figure 3: The completion of $G$ obtained by superimposing $G^\sigma$ and its suspended dual $G^{\sigma^*}$ (the latter depicted with dotted edges).
  • Figure 4: Illustration of the proof of Lemma \ref{['lem_deg_leq_5']}.
  • Figure 5: The two possibilities discussed in Case 1 and 2.

Theorems & Definitions (16)

  • Definition 1
  • Definition 2
  • Lemma 3: Kant1992F04LSFPG
  • Corollary 4
  • Definition 5
  • Lemma 6: Alam et al. Alam2015
  • Definition 7
  • Definition 8
  • Lemma 9
  • proof
  • ...and 6 more