2026-07-23T05:31:04Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:20077 2024-05-31T13:11:12Z ddc:004 open_access

Toward Grünbaum’s Conjecture Ortlieb, Christian Schmidt, Jens M. Planar graph spanning tree maximum degree Schnyder wood 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Christian Ortlieb and Jens M. Schmidt 2024 Is Part Of LIPIcs, Volume 294, 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SWAT.2024.37 urn:nbn:de:0030-drops-200777 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.37 eng https://creativecommons.org/licenses/by/4.0/legalcode