**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. |