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Toward Grünbaum’s Conjecture for 4-Connected Graphs

Author Christian Ortlieb

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Author Details

Christian Ortlieb
  • 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. Toward Grünbaum’s Conjecture for 4-Connected Graphs. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 77:1-77:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.MFCS.2024.77

Abstract

Given a spanning tree T of a 3-connected planar graph G, the co-tree of T is the spanning tree of the dual graph G^* given by the duals of the edges that are not in T. Grünbaum conjectured in 1970 that there is such a spanning tree T such that T and its co-tree both have maximum degree at most 3.
In 2014, Biedl proved that there is a spanning tree T such that T and its co-tree have maximum degree at most 5. Using structural insights into Schnyder woods, Schmidt and the author recently improved this bound on the maximum degree to 4. In this paper, we prove that in a 4-connected planar graph there exists a spanning tree T of maximum degree at most 3 such its co-tree has maximum degree at most 4. This almost solves Grünbaum’s conjecture for 4-connected graphs.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
Keywords
  • 4-connected planar graph
  • spanning tree
  • maximum degree
  • Schnyder wood
  • Grünbaum

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