13 Members of a new club ,meet each day for lunch at a round table. They decide to sit such that every memher has different neighbours at each lunch.How many days can this arrangement last?
By using normal combinatorics we can find that number of days is six. But I have no idea how to do it using graph theory.
Draw the thirteen vertices in a circle. You can draw six different graphs on these vertices by connecting each vertex to the on next to it, skip one, skip two, etc. Each graph is a cycle graph and indicates the order around the table for one day if you follow the cycle.