Question

Suppose we have 2n people, some of which are related to some of the others. We might want to split them into groups of two, so that the two people in a group are related (if this is possible) Expressing this as a graph problem, suppose we have an undirected graph G-(WB). A pairing is a set P C E of edges such that for all (u,v),(x,y) є P, the nodes u,v,z, y are all different. In other words, no two edges in P have a node in common. A complete pairing is a pairing P that uses all the graph's nodes, that is, a pairing for which U(ue)EP[u,v-V. For example, in the graph shown below, one possible complete pairing is (a, b), (d, e), (c,f)b, but there is no complete pairing that contains the edge (b, c) Use mathematical induction to show how to construct a graph with 2n nodes and n2 edges such that the graph has exactly one, unique, complete pairing.

Answer 1

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