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Hamiltonian Properties of Generalized Halin Graphs

Qureshi, Ahmad Mehmood Hamiltonian Properties of Generalized Halin Graphs. PhD thesis, Govt. College University, Lahore.

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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. Halin graphs were introduced by R. Halin [16] as a class of minimally 3-connected planar graphs. They also possess interesting Hamiltonian properties. They are 1-Hamiltonian, i.e., they are Hamiltonian and remain so after the removal of any single vertex, as Bondy showed (see [23]). Moreover, Barefoot proved that they are Hamiltonian connected, i.e., they admit a Hamiltonian path be­tween every pair of vertices [1]. Bondy and Lovasz [6] and, independently, Skowronska [33] proved that Halin graphs on n vertices are almost pancyclic, more precisely they contain cycles of all lengths l (3 l n) except possibly for a single even length. Also, they showed that Halin graphs on n vertices whose vertices of degree 3 are all on the outer cycle C are pancyclic, i.e., they must contain cycles of all lengths from 3 to n. In this thesis, we dene classes of generalized Halin graphs, called k-Halin graphs, and investigate their Hamiltonian properties. In chapter 4, we dene k-Halin graph in the following way. A 2-connected planar graph G without vertices of degree 2, possessing a cycle C such that (i) all vertices of C have degree 3 in G, and (ii) G C is connected and has at most k cycles is called a k-Halin graph. A 0-Halin graph, thus, is a usual Halin graph. Moreover, the class of k-Halin graphs is contained in the class of (k + 1)-Halin graphs (k 0). We shall see that, the Hamiltonicity of k-Halin graphs steadily decreases as k increases. Indeed, a 1-Halin graph is still Hamiltonian, but not Hamiltonian con­nected, a 2-Halin graph is not necessarily Hamiltonian but still traceable, while a 3-Halin graph is not even necessarily traceable. The property of being 1-Hamiltonian, Hamiltonian connected or almost pancyclic is not preserved, even by 1-Halin graphs. However, Bondy and Lovaszresult about the pancyclicity of Halin graphs with no inner vertex of degree 3 remains true even for 3-Halin graphs. The property of being Hamiltonian persists, however, for large values of k in cubic 3-connected k-Halin graphs. In chapter 5, it will be shown that every cubic 3-connected 14-Halin graph is Hamiltonian. A variant of the famous example of Tutte [37] from 1946 which rst demonstrated that cubic 3-connected planar graphs may not be Hamiltonian, is a 21-Halin graphs. The cubic 3-connected planar non-Hamiltonian graph of Lederberg [21], Bosak [7] and Barnette, which has smallest order, is 53-Halin. The sharpness of our result is proved by showing that there exist non-Hamiltonian cubic 3-connected 15-Halin graphs.

Item Type:Thesis (PhD)
Subjects:Physical Sciences (f) > Mathematics(f5)
ID Code:2665
Deposited By:Mr. Javed Memon
Deposited On:22 Jul 2009 11:17
Last Modified:25 Jul 2009 09:14

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