Hamiltonian Properties of Generalized Halin Graphs

Abstract

Abstract.A Halin graph is a graph H = T ∪ C, where T is a tree with no vertex of degree two, and C is a cycle connecting the end-vertices of T in the cyclic order determined by a plane embedding of T. In this paper, we define classes of generalized Halin graphs, called k-Halin graphs, and investigate their Hamiltonian properties.

Hamiltonian properties of generalized pyramids

“…Other generalizations of Halin graphs and investigations of their hamiltonian properties have already been made by Skowrońska 6, Skowrońska and Sysło 7, Skupień 8, and Malik, Qureshi and Zamfirescu 5.…”

Hamiltonicity of Cubic 3-Connected k-Halin Graphs

“…Halin graphs belong to the family of all planar, 3-connected graphs and possess strong hamiltonian properties. We presented in [13] the following generalization of Halin graphs.…”

“…As expected, as k increases, the hamiltonicity of k-Halin graphs steadily decreases. It was proved in [13] that a 1-Halin graph is still hamiltonian, but may not be hamiltonian connected, a 2-Halin graph is not necessarily hamiltonian but still traceable, while a 3-Halin graph is not even necessarily traceable. However, like Halin graphs, those 3-Halin graphs, in which vertices of degree 3 are on the outer cycle only, are pancyclic.…”