199 research outputs found
Bipartite 2-factorizations of complete multipartite graphs
It is shown that if K is any regular complete multipartite graph of even degree, and F is any bipartite 2-factor of K, then there exists a factorization of K into F; except that there is no factorization of K into F when F is the union of two disjoint 6-cycles
On the Hamilton-Waterloo problem for bipartite 2-factors
Given two 2-regular graphs F1 and F2, both of order n, the Hamilton-Waterloo Problem for F1 and F2 asks for a factorization of the complete graph Kn into a1 copies of F1, a2 copies of F2, and a 1-factor if n is even, for all nonnegative integers a1 and a2 satisfying a1+a2=?n-12?. We settle the Hamilton-Waterloo Problem for all bipartite 2-regular graphs F1 and F2 where F1 can be obtained from F2 by replacing each cycle with a bipartite 2-regular graph of the same order
On the minisymposium problem
The generalized Oberwolfach problem asks for a factorization of the complete
graph into prescribed -factors and at most a -factor. When all
-factors are pairwise isomorphic and is odd, we have the classic
Oberwolfach problem, which was originally stated as a seating problem: given
attendees at a conference with circular tables such that the th
table seats people and , find a seating
arrangement over the days of the conference, so that every
person sits next to each other person exactly once.
In this paper we introduce the related {\em minisymposium problem}, which
requires a solution to the generalized Oberwolfach problem on vertices that
contains a subsystem on vertices. That is, the decomposition restricted to
the required vertices is a solution to the generalized Oberwolfach problem
on vertices. In the seating context above, the larger conference contains a
minisymposium of participants, and we also require that pairs of these
participants be seated next to each other for
of the days.
When the cycles are as long as possible, i.e.\ , and , a flexible
method of Hilton and Johnson provides a solution. We use this result to provide
further solutions when and all cycle lengths are
even. In addition, we provide extensive results in the case where all cycle
lengths are equal to , solving all cases when , except possibly
when is odd and is even.Comment: 25 page
Oberwolfach rectangular table negotiation problem
AbstractWe completely solve certain case of a “two delegation negotiation” version of the Oberwolfach problem, which can be stated as follows. Let H(k,3) be a bipartite graph with bipartition X={x1,x2,…,xk},Y={y1,y2,…,yk} and edges x1y1,x1y2,xkyk−1,xkyk, and xiyi−1,xiyi,xiyi+1 for i=2,3,…,k−1. We completely characterize all complete bipartite graphs Kn,n that can be factorized into factors isomorphic to G=mH(k,3), where k is odd and mH(k,3) is the graph consisting of m disjoint copies of H(k,3)
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