Oberwolfach problem

The Oberwolfach problem is an unsolved problem in mathematics that may be formulated either as a problem scheduling seating assignments for diners, or more abstractly as a problem in graph theory, on the edge cycle covers of complete graphs. It is named after the Mathematical Research Institute of Oberwolfach, where the problem was posed in 1967 by Gerhard Ringel.

In conferences held at Oberwolfach, it is the custom for the participants to dine together in a room with circular tables, not all the same size, and with assigned seating that rearranges the participants from meal to meal. The Oberwolfach problem asks how to make a seating chart for a given set of tables so that all tables are full at each meal and all pairs of conference participants are seated next to each other exactly once. An instance of the problem can be denoted as ${\displaystyle OP(x,y,z,\dots )}$ where ${\displaystyle x,y,z,\dots }$ are the given table sizes. Alternatively, when some table sizes are repeated, they may be denoted using exponential notation; for instance, ${\displaystyle OP(5^{3})}$ describes an instance with three tables of size five.

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