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Toward Grünbaum’s Conjecture

Authors Christian Ortlieb, Jens M. Schmidt

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Author Details

Christian Ortlieb
  • Institute of Computer Science, University of Rostock, Germany
Jens M. Schmidt
  • Institute of Computer Science, University of Rostock, Germany

Funding

This research is supported by the grant SCHM 3186/2-1 (401348462) from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation).

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Christian Ortlieb and Jens M. Schmidt. Toward Grünbaum’s Conjecture. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 37:1-37:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SWAT.2024.37

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. This tree can be computed in linear time.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
Keywords
  • Planar graph
  • spanning tree
  • maximum degree
  • Schnyder wood

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