17 research outputs found
Two Problems of Gerhard Ringel
Gerhard Ringel was an Austrian Mathematician, and is regarded as one of the most influential graph theorists of the twentieth century. This work deals with two problems that arose from Ringel\u27s research: the Hamilton-Waterloo Problem, and the problem of R-Sequences.
The Hamilton-Waterloo Problem (HWP) in the case of Cm-factors and Cn-factors asks whether Kv, where v is odd (or Kv-F, where F is a 1-factor and v is even), can be decomposed into r copies of a 2-factor made entirely of m-cycles and s copies of a 2-factor made entirely of n-cycles. Chapter 1 gives some general constructions for such decompositions and apply them to the case where m=3 and n=3x. This problem is settle for odd v, except for a finite number of x values. When v is even, significant progress is made on the problem, although open cases are left. In particular, the difficult case of v even and s=1 is left open for many situations.
Chapter 2 generalizes the Hamilton-Waterloo Problem to complete equipartite graphs K(n:m) and shows that K(xyzw:m) can be decomposed into s copies of a 2-factor consisting of cycles of length xzm and r copies of a 2-factor consisting of cycles of length yzm, whenever m is odd, s,r≠1, gcd(x,z)=gcd(y,z)=1 and xyz≠0 (mod 4). Some more general constructions are given for the case when the cycles in a given two factor may have different lengths. These constructions are used to find solutions to the Hamilton-Waterloo problem for complete graphs.
Chapter 3 completes the proof of the Friedlander, Gordon and Miller Conjecture that every finite abelian group whose Sylow 2-subgroup either is trivial or both non-trivial and non-cyclic is R-sequenceable. This settles a question of Ringel for abelian groups
A Generalization of the Hamilton-Waterloo Problem on Complete Equipartite Graphs
The Hamilton-Waterloo problem asks for which and the complete graph
can be decomposed into copies of a given 2-factor and
copies of a given 2-factor (and one copy of a 1-factor if is even).
In this paper we generalize the problem to complete equipartite graphs
and show that can be decomposed into copies of a
2-factor consisting of cycles of length ; and copies of a 2-factor
consisting of cycles of length , whenever is odd, ,
and . We also give some more general
constructions where the cycles in a given two factor may have different
lengths. We use these constructions to find solutions to the Hamilton-Waterloo
problem for complete graphs
The Hamilton-Waterloo Problem with even cycle lengths
The Hamilton-Waterloo Problem HWP asks for a
2-factorization of the complete graph or , the complete graph with
the edges of a 1-factor removed, into -factors and
-factors, where . In the case that and are both
even, the problem has been solved except possibly when
or when and are both odd, in which case necessarily . In this paper, we develop a new construction that creates
factorizations with larger cycles from existing factorizations under certain
conditions. This construction enables us to show that there is a solution to
HWP for odd and whenever the obvious
necessary conditions hold, except possibly if ; and
; ; or . This result almost completely
settles the existence problem for even cycles, other than the possible
exceptions noted above
On the Hamilton-Waterloo problem: the case of two cycles sizes of different parity
The Hamilton-Waterloo problem asks for a decomposition of the complete graph of order v into r copies of a 2-factor F1 and s copies of a 2-factor F2 such that r+s = v−1 2 . If F1 consists of m-cycles and F2 consists of n cycles, we say that a solution to (m, n)- HWP(v; r, s) exists. The goal is to find a decomposition for every possible pair (r, s). In this paper, we show that for odd x and y, there is a solution to (2kx, y)-HWP(vm; r, s) if gcd(x, y) ≥ 3, m ≥ 3, and both x and y divide v, except possibly when 1 ∈ {r, s}
Constructing uniform 2-factorizations via row-sum matrices: solutions to the Hamilton-Waterloo problem
In this paper, we formally introduce the concept of a row-sum matrix over an
arbitrary group . When is cyclic, these types of matrices have been
widely used to build uniform 2-factorizations of small Cayley graphs (or,
Cayley subgraphs of blown-up cycles), which themselves factorize complete
(equipartite) graphs.
Here, we construct row-sum matrices over a class of non-abelian groups, the
generalized dihedral groups, and we use them to construct uniform
-factorizations that solve infinitely many open cases of the
Hamilton-Waterloo problem, thus filling up large parts of the gaps in the
spectrum of orders for which such factorizations are known to exist