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.
@InProceedings{ortlieb_et_al:LIPIcs.SWAT.2024.37, author = {Ortlieb, Christian and Schmidt, Jens M.}, title = {{Toward Gr\"{u}nbaum’s Conjecture}}, booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)}, pages = {37:1--37:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-318-8}, ISSN = {1868-8969}, year = {2024}, volume = {294}, editor = {Bodlaender, Hans L.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.37}, URN = {urn:nbn:de:0030-drops-200777}, doi = {10.4230/LIPIcs.SWAT.2024.37}, annote = {Keywords: Planar graph, spanning tree, maximum degree, Schnyder wood} }
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