On the minisymposium problem

The generalized Oberwolfach problem asks for a factorization of the complete graph K-v 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 ai i people and Sigma(t)(i=1) a(i) = v, find a seating arrangement over the v-1/2 days of the conference, so that every person sits next to each other person exactly once.In this paper we introduce the related 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 (sic)m-1/2(sic) of the days.When the cycles are as long as possible, i.e. v, m and v - m, a flexible method of Hilton and Johnson provides a solution. We use this result to provide further solutions when v equivalent to m equivalent to 2 (mod 4) 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 m | v, except possibly when k is odd and v is even.