17 research outputs found
Brick assignments and homogeneously almost self-complementary graphs
AbstractA graph is called almost self-complementary if it is isomorphic to the graph obtained from its complement by removing a 1-factor. In this paper, we study a special class of vertex-transitive almost self-complementary graphs called homogeneously almost self-complementary. These graphs occur as factors of symmetric index-2 homogeneous factorizations of the “cocktail party graphs” K2n−nK2. We construct several infinite families of homogeneously almost self-complementary graphs, study their structure, and prove several classification results, including the characterization of all integers n of the form n=pr and n=2p with p prime for which there exists a homogeneously almost self-complementary graph on 2n vertices
The directed Oberwolfach problem with variable cycle lengths: a recursive construction
The directed Oberwolfach problem OP asks whether the
complete symmetric digraph , assuming , admits a
decomposition into spanning subdigraphs, each a disjoint union of directed
cycles of lengths . We hereby describe a method for
constructing a solution to OP given a solution to
OP, for some , if certain conditions on
are satisfied. This approach enables us to extend a solution
for OP into a solution for
OP, as well as into a solution for
OP, where denotes copies of 2, provided is sufficiently large.
In particular, our recursive construction allows us to effectively address
the two-table directed Oberwolfach problem. We show that OP has
a solution for all , with a definite exception of
and a possible exception in the case that , is even,
and . It has been shown previously that OP has
a solution if is odd, and that OP has a solution if and
only if .
In addition to solving many other cases of OP, we show that when , OP has a solution if and
only if
Using edge cuts to find Euler tours and Euler families in hypergraphs
An Euler tour in a hypergraph is a closed walk that traverses each edge of
the hypergraph exactly once, while an Euler family is a family of closed walks
that jointly traverse each edge exactly once and cannot be concatenated. In
this paper, we show how the problem of existence of an Euler tour (family) in a
hypergraph can be reduced to the analogous problem in some smaller
hypergraphs that are derived from using an edge cut of . In the process,
new techniques of edge cut assignments and collapsed hypergraphs are
introduced. Moreover, we describe algorithms based on these characterizations
that determine whether or not a hypergraph admits an Euler tour (family), and
can also construct an Euler tour (family) if it exists
Cycle decompositions IV: complete directed graphs and fixed length directed cycles
We establish necessary and sufficient conditions for decomposing the complete symmetric digraph of order n into directed cycles of length m; where 2≼m≼n