A note on a conjecture concerning tree-partitioning 3-regular graphs

T. Bohme, T. Bohme, Haitze J. Broersma, Hilde Tuinstra

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If $G$ is a 4-connected maximal planar graph, then $G$ is Hamiltonian (by a theorem of Whitney), implying that its dual graph $G^*$ is a cyclically 4-edge connected 3-regular planar graph admitting a partition of the vertex set into two parts, each inducing a tree in $G^*$, a so-called tree-partition. It is a natural question whether each cyclically 4-edge connected 3-regular graph admits such a tree-partition. This was conjectured by Jaeger, and recently independently by the first author. The main result of this note shows that each connected 3-regular graph on $n$ vertices admits a partition of the vertex set into two sets such that precisely $n/2+2$ edges have end vertices in each set. This is a necessary condition for having a tree-partition. We also show that not all cyclically 3-edge connected 3-regular (planar) graphs admit a tree-partition, and present the smallest counterexamples.
Original languageUndefined
Place of PublicationEnschede
PublisherUniversity of Twente
Number of pages7
ISBN (Print)0169-2690
Publication statusPublished - 1998

Publication series

NameMemorandum / University of Twente, Faculty of Applied Mathematics, ISSN 0921-1969 ; no. 1427
PublisherUniversiteit Twente


  • MSC-05C35
  • MSC-05C38
  • IR-30611
  • EWI-3247
  • METIS-141252
  • MSC-05C45

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