Lovász-Plummer conjecture on Halin graphs

A Halin graph, defined by Halin [3], is a plane graph H = T ∪ C such that T is a spanning tree of H with no vertices of degree 2 where |T | ≥ 4 and C is a cycle whose vertex set is the set of leaves of T . In his work, as an example of a class of edge-minimal 3-connected plane graphs, Halin constructed this family of plane graphs, which have many interesting properties. Lovász and Plummer [5] named such a graph a Halin graph. Note that every tree with no vertices of degree 2 contains two leaves that share the same parent. By the planarity, in a Halin graph H = T ∪ C, there are two leaves of T sharing the same parent which are consecutive on C, so H contains a triangle. Consequently, no Halin graph is bipartite. After Halin constructed such a class, it has been proved that Halin graphs have interesting properties. For example, Bondy [2] proved that every Halin graph is Hamiltonian. Moreover, Barefoot [1] proved the following result stronger than that of Bondy [2]; every Halin graph is Hamilton-connected. Since Halin graphs have nice properties, it is natural to study graphs containing Halin graph as a spanning subgraph. We call such a spanning subgraph a spanning Halin subgraph. Horton, Parker and Borie [4] proved that deciding whether a graph has a spanning Halin subgraph is NP-complete. So, it is feasible to consider special classes of graphs. Lovász and Plummer [5] conjectured that every 4-connected plane triangulation has a spanning Halin subgraph. In this talk, we disprove their conjecture by constructing plane triangulations with no spanning Halin subgraph. Furthermore, we show that even if we assume “5-connectedness”, the conjecture is wrong.