Abstract
To
determine whether or not a given graph has a hamiltonian cycle
is much harder than deciding whether it is Eulerian, and no
algorithmically useful characterization of hamiltonian graphs is
known, although several necessary conditions and many sufficient
conditions (see [6]) have been discovered. In fact, it is known
that determining whether there are hamiltonian paths or cycles
in arbitrary graphs is N Pcomplete. The interested reader is
referred in particular to the surveys of Berge ([5], Chapter
10), Bondy and Murty ([10], Chapters 4 and 9), J. C. Bermond
[6], Flandrin, Faudree and Ryjacstek [21] and R. Gould [27].
Hamiltonicity in special classes of graphs is a major area of
graph theory and a lot of graph theorists have studied it. One
special class of graphs whose hamiltonicity has been studied is
that of Toeplitz graphs, introduced by van Dal et al. [13] in
1996. This study was continued by C. Heuberger [32] in 2002. The
Toeplitz graphs investigated in [13] and [32] were all
undirected. We intend to extend here this study to the directed
case. A Toeplitz matrix, named after Otto Toeplitz, is a square
matrix (n n) which has constant values along all diagonals
parallel to the main diagonal. Thus, Toeplitz matrices are dened
by 2n1 numbers. Toeplitz matrices have uses in dierent areas in
pure and applied mathematics, and also in computer science. For
example, they are closely connected with Fourier series, they
often appear when dierential or integral equations are
discretized, they arise in physical dataprocessing
applications, in the theories of orthogonal polynomials,
stationary processes, and moment problems; see Heinig and Rost
[31]. For other references on Toeplitz matrices see [26], [28]
and [33].
A
special case of a Toeplitz matrix is a circulant matrix, where
each row is rotated one element to the right relative to the
preceding row. Circulant matrices and their properties have been
studied in [14] and [28]. In numerical analysis circulant
matrices are important because they are diagonalized by a
discrete Fourier transform, and hence linear equations that
contain them may be quickly solved using a fast Fourier
transform. These matrices are also very useful in digital image
processing. A directed or undirected graph whose adjacency
matrix is circulant is called circulant. Circulant graphs and
their properties such as connectivity, hamiltonicity,
bipartiteness, planarity and colourability have been studied by
several authors (see [8], [11], [15], [25], [35], [38], [41] and
[24]). In particular, the conjecture of Boesch and Tindell [8],
that all undirected connected circulant graphs are hamiltonian,
was proved by Burkard and Sandholzer [11]. A directed or
undirected Toeplitz graph is dened by a Toeplitz adjacency
matrix. The properties of Toeplitz graphs; such as bipartiteness,
planarity and colourability, have been studied in [18], [19],
[20]. Hamiltonian properties of undirected Toeplitz graphs have
been studied in [13] and [32]. For arbitrary digraphs the
hamiltonian path and cycle problems are also very difcult and
both are N Pcomplete (see, e.g. the book [22] by Garey and
Johnson). It is worthwhile mentioning that the hamiltonian cycle
and path problems are N Pcomplete even for some special classes
of digraphs. Garey, Johnson and Tarjan shows [23] that the
problem remains N Pcomplete even for planar 3regular digraphs.
Some powerful necessary conditions, due to Gutin and Yeo [10],
are considered for a digraph to be hamiltonian. For information
on hamiltonian and traceable digraphs, see e.g. the survey [2]
and [3] by BangJensen and Gutin, [9] by Bondy, [29] by Gutin
and [39] by Volkmann.
In
this thesis, we investigate the hamiltonicity of directed
Toeplitz graphs. The main purpose of this thesis is to oer
sucient conditions for the existence of hamiltonian paths and
cycles in directed Toeplitz graphs, which we will discuss in
Chapters 3 and 4. The main diagonal of an (n n) Toeplitz
adjacency matrix will be labeled 0 and it contains only zeros.
The n 1 distinct diagonals above the main diagonal will be
labeled 12n 1 and those under the main diagonal will also be
labeled 12n 1. Let s1 s2sk be the upper diagonals containing
ones and t1 t2tl be the lower diagonals containing ones, such
that 0 < s1 < s2 < < sk < n and 0 < t1 < t2
< < tl < n. Then, the
corresponding directed Toeplitz graph will be denoted by Tns1
s2sk; t1t2tl. That is, Tns1 s2 sk; t1t2tlis the graph with
vertex set 12n, in which the edge (ij), 1 · i <j · n, occurs if
and only if j i = sp or i j = tq for some p andq(1 · p · k, 1 ·
q· l). In Chapter 1 we describe some basic ideas, terminology
and results about graphs and digraphs. Further we discuss
adjacency matrices, Toeplitz matrices, which we will encounter
in the following chapters. In Chapter 2 we discuss hamiltonian
graphs and add a brief historical note. We then discuss
undirected Toeplitz graph, and nally mention some known results
on hamiltonicity of undirected Toeplitz graphs found by van Dal
et al. [13] and C. Heuberger [32].
Since all
graphs in the main part of the thesis (Chapters 3 and 4) will be
directed, we shall omit mentioning it in these chapters. We shall
consider here just graphs without loops, because loops play no role
in hamiltonicity investigations. Thus, unless otherwise mentioned,
in Chapters 3 and 4, by a graph we always mean a nite simple
digraph.
In
Chapter 3, for k = l = 1 we obtain a characterization of cycles
among directed Toeplitz graphs, and another result similar to
Theorem 10 in [13]. Directed Toeplitz graphs with s1 = 1 (or t1 = 1)
are obviously traceable. If we ask moreover that s2 = 2, we see that
the hamiltonicity of Tn12; t1depends upon the parity of t1 and n.
Further in the same Chapter, we require s3 = 3 and succeed to prove
the hamiltonicity of Tn123; t1for all t1 and n.
In
Chapter 4 we present a few results on Toeplitz graphs with s1 = t1 =
1 and s2 = 3. They will often depend upon the parity of n.
Chapter 5
contains some concluding remarks.
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