Let Gh be the collection of all graphs with the vertex set V = {1, 2,...
Let Gh be the collection of all graphs with the vertex set V = {1, 2, 3, 4, 5, 6, 7, 8}. Suppose we are given a list of 32 graphs G1, G2, . . . G32, each in Gh.
(a) The following argument is wrong. Identify the error.
There are 
= 28 two-element subsets of V . Given any graph G ∈ Gh, each edge e
∈ E(G) is a two-element subset of V . So there are 28 different
graphs possible, and |Gh| = 28. Since our list includes 32
different graphs, and 32 > 28, by the pigeonhole principle at
least two of the graphs in the list must be the same. (That is, Gi
= Gj for some 1 ≤ i < j ≤ k.)
(b) How many graphs do we really need in the list to assure that at least two are the same?
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