Trees and hamiltonicity

Various classes of graphs possessing certain Hamiltonian properties are presented. In each class, either the graphs in question emanated from a tree, the proof that the graphs possess some specified Hamiltonian property makes use of trees or the Hamiltonian property itself involves trees. The 3-path graph P3{G) of a connected graph G of order 3 or more has the set of all 3-paths (paths of order 3) of G as its vertex set and two vertices of P3(G) are adjacent if they have a 2-path (an edge) in common. We provide sufficient conditions for the 3-path graph of a tree to be Hamiltonian or Hamiltonian-connected. For an integer k ≥ 2, a connected graph G of order at least k + 1 is k-tree-connected if for every set S of k distinct vertices of G, there exists a spanning tree T of G whose set of end-vertices is S. Sufficient conditions are presented for the 3-path graph of a tree to be 3-tree-connected and for a graph, in general, to be k-tree-connected. Furthermore, it is shown for each integer k ≥ 3 that the kth power of every connected graph of order k or more is (k - l)-tree-connected. © Copyright 2018, Charles Babbage Research Centre. All rights reserved.