19 research outputs found
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
A complete solution to the infinite Oberwolfach problem
Let be a -regular graph of order . The Oberwolfach problem,
, asks for a -factorization of the complete graph on vertices in
which each -factor is isomorphic to . In this paper, we give a complete
solution to the Oberwolfach problem over infinite complete graphs, proving the
existence of solutions that are regular under the action of a given involution
free group . We will also consider the same problem in the more general
contest of graphs that are spanning subgraphs of an infinite complete graph
and we provide a solution when is locally finite. Moreover, we
characterize the infinite subgraphs of such that there exists a
solution to containing a solution to
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}
A constructive solution to the Oberwolfach Problem with a large cycle
For every -regular graph of order , the Oberwolfach problem
asks whether there is a -factorization of ( odd) or minus a
-factor ( even) into copies of . Posed by Ringel in 1967 and
extensively studied ever since, this problem is still open. In this paper we
construct solutions to whenever contains a cycle of length greater
than an explicit lower bound. Our constructions combine the
amalgamation-detachment technique with methods aimed at building
-factorizations with an automorphism group having a nearly-regular action on
the vertex-set.Comment: 11 page
Infinitely many cyclic solutions to the Hamilton-Waterloo problem with odd length cycles
It is conjectured that for every pair of odd integers greater than
2 with , there exists a cyclic two-factorization of
having exactly factors of type and all the
others of type . The authors prove the conjecture in the affirmative
when and .Comment: 31 page