The generalized Oberwolfach problem asks for a factorization of the complete
graph Kvβ into prescribed 2-factors and at most a 1-factor. When all
2-factors are pairwise isomorphic and v is odd, we have the classic
Oberwolfach problem, which was originally stated as a seating problem: given
v attendees at a conference with t circular tables such that the ith
table seats aiβ people and βi=1tβaiβ=v, find a seating
arrangement over the 2vβ1β 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 v vertices that
contains a subsystem on m vertices. That is, the decomposition restricted to
the required m vertices is a solution to the generalized Oberwolfach problem
on m vertices. In the seating context above, the larger conference contains a
minisymposium of m participants, and we also require that pairs of these m
participants be seated next to each other for
β2mβ1ββ of the days.
When the cycles are as long as possible, i.e.\ v, m and vβm, a flexible
method of Hilton and Johnson provides a solution. We use this result to provide
further solutions when vβ‘mβ‘2(mod4) and all cycle lengths are
even. In addition, we provide extensive results in the case where all cycle
lengths are equal to k, solving all cases when mβ£v, except possibly
when k is odd and v is even.Comment: 25 page