18 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
On some third parts of nearly complete digraphs
AbstractFor the complete digraph DKn with n⩾3, its half as well as its third (or near-third) part, both non-self-converse, are exhibited. A backtracking method for generating a tth part of a digraph is sketched. It is proved that some self-converse digraphs are not among the near-third parts of the complete digraph DK5 in four of the six possible cases. For n=9+6k,k=0,1,…, a third part D of DKn is found such that D is a self-converse oriented graph and all D-decompositions of DKn have trivial automorphism group
Enumerating Hamiltonian Cycles in A 2-connected Regular Graph Using Planar Cycle Bases
Planar fundamental cycle basis belong to a 2-connected simple graph is used for
enumerating Hamiltonian cycles contained in the graph. This is because a fun-
damental cycle basis is easily constructed. Planar basis is chosen since it has a
weighted induced graph whose values are limited to 1. Hence making it is possible
to be used in the Hamiltonian cycle enumeration procedures efficiently. In this
paper a Hamiltonian cycle enumeration scheme is obtained through two stages.
Firstly, i cycles out of m bases cycles are determined using an appropriate con-
structed constraint. Secondly, to search all Hamiltonian cycles which are formed
by the combination of i basis cycles obtained in the first stage efficiently. This ef-
ficiency is achieved through the generation of a class of objects consisting of Ill-bit
binary strings which is a representation of i cycle combinations between m cycle
basis cycle
The Hamilton–Waterloo problem: the case of Hamilton cycles and triangle-factors
AbstractWe discuss a special case of the Hamilton–Waterloo problem in which a 2-factorization of Kn is sought consisting of 2-factors of two kinds: Hamiltonian cycles, and triangle-factors. We determine completely the spectrum of solutions for several infinite classes of orders n
On Hamilton decompositions of infinite circulant graphs
The natural infinite analogue of a (finite) Hamilton cycle is a two-way-infinite Hamilton path (connected spanning 2-valent subgraph).
Although it is known that every connected 2k-valent infinite circulant graph has a two-way-infinite Hamilton path, there exist many such graphs that do not have a decomposition into k edge-disjoint two-way-infinite Hamilton paths. This contrasts with the finite case where it is conjectured that every 2k-valent connected circulant graph has a decomposition into k edge-disjoint Hamilton cycles. We settle the problem of decomposing 2k-valent infinite circulant graphs into k edge-disjoint two-way-infinite Hamilton paths for k=2, in many cases when k=3, and in many other cases including where the connection set is ±{1,2,...,k} or ±{1,2,...,k - 1, 1,2,...,k + 1}
On the longest path in a recursively partitionable graph
A connected graph with order is said to be recursively arbitrarily partitionable (R-AP for short) if either it is isomorphic to , or for every sequence of positive integers summing up to there exists a partition of such that each induces a connected R-AP subgraph of on vertices. Since previous investigations, it is believed that a R-AP graph should be 'almost traceable' somehow. We first show that the longest path of a R-AP graph on vertices is not constantly lower than for every . This is done by exhibiting a graph family such that, for every positive constant , there is a R-AP graph in that has arbitrary order and whose longest path has order . We then investigate the largest positive constant such that every R-AP graph on vertices has its longest path passing through vertices. In particular, we show that . This result holds for R-AP graphs with arbitrary connectivity