eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-05-31
37:1
37:17
10.4230/LIPIcs.SWAT.2024.37
article
Toward Grünbaum’s Conjecture
Ortlieb, Christian
1
Schmidt, Jens M.
1
https://orcid.org/0000-0003-3032-4834
Institute of Computer Science, University of Rostock, Germany
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol294-swat2024/LIPIcs.SWAT.2024.37/LIPIcs.SWAT.2024.37.pdf
Planar graph
spanning tree
maximum degree
Schnyder wood