8 research outputs found
A generalization of the problem of Mariusz Meszka
Mariusz Meszka has conjectured that given a prime p=2n+1 and a list L
containing n positive integers not exceeding n there exists a near 1-factor in
K_p whose list of edge-lengths is L. In this paper we propose a generalization
of this problem to the case in which p is an odd integer not necessarily prime.
In particular, we give a necessary condition for the existence of such a near
1-factor for any odd integer p. We show that this condition is also sufficient
for any list L whose underlying set S has size 1, 2, or n. Then we prove that
the conjecture is true if S={1,2,t} for any positive integer t not coprime with
the order p of the complete graph. Also, we give partial results when t and p
are coprime. Finally, we present a complete solution for t<12.Comment: 15 page
The seating couple problem in even case
In this paper we consider the seating couple problem with an even number of
seats, which, using graph theory terminology, can be stated as follows. Given a
positive even integer and a list containing positive integers
not exceeding , is it always possible to find a perfect matching of
whose list of edge-lengths is ? Up to now a (non-constructive) solution is
known only when all the edge-lengths are coprime with . In this paper we
firstly present some necessary conditions for the existence of a solution.
Then, we give a complete constructive solution when the list consists of one or
two distinct elements, and when the list consists of consecutive integers
, each one appearing with the same multiplicity. Finally, we
propose a conjecture and some open problems.Comment: 16 page
A problem on partial sums in abelian groups
In this paper we propose a conjecture concerning partial sums of an arbitrary
finite subset of an abelian group, that naturally arises investigating simple
Heffter systems. Then, we show its connection with related open problems and we
present some results about the validity of these conjectures
Chromatic numbers of Cayley graphs of abelian groups: A matrix method
In this paper, we take a modest first step towards a systematic study of
chromatic numbers of Cayley graphs on abelian groups. We lose little when we
consider these graphs only when they are connected and of finite degree. As in
the work of Heuberger and others, in such cases the graph can be represented by
an integer matrix, where we call the dimension and the
rank. Adding or subtracting rows produces a graph homomorphism to a graph with
a matrix of smaller dimension, thereby giving an upper bound on the chromatic
number of the original graph. In this article we develop the foundations of
this method. In a series of follow-up articles using this method, we completely
determine the chromatic number in cases with small dimension and rank; prove a
generalization of Zhu's theorem on the chromatic number of -valent integer
distance graphs; and provide an alternate proof of Payan's theorem that a
cube-like graph cannot have chromatic number 3.Comment: 17 page
Starter sequences: generalizations and applications
In this thesis we introduce new types of starter sequences, pseudo-starter sequences,
starter-labellings, and generalized (extended) starter sequences. We apply these new
sequences to graph labeling. All the necessary conditions for the existence of starter,
pseudo-starter, extended, m-fold, excess, and generalized (extended) starter sequences
are determined, and some of these conditions are shown to be sufficient. The relationship between starter sequences and graph labellings is introduced. Moreover, the starter-labeling and the minimum hooked starter-labeling of paths, cycles, and k-
windmills are investigated. We show that all paths, cycles, and k-windmills can be
starter-labelled or minimum starter-labelled
A Generalization of the Problem of Mariusz Meszka
Mariusz Meszka has conjectured that given a prime p = 2n + 1 and a list L
containing n positive integers not exceeding n there exists a near 1-factor in K_p whose list of edge-lengths is L. In this paper we propose a generalization of this problem to the case in which p is an odd integer not necessarily prime. In particular, we give a necessary condition for the existence of such a near 1-factor for any odd integer p.
We show that this condition is also sufficient for any list L whose underlying set S has size 1, 2, or n. Then we prove that the conjecture is true if S = {1, 2, t} for any
positive integer t not coprime with the order p of the complete graph. Also, we give partial results when t and p are coprime. Finally, we present a complete solution for t 64 11