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If we have the complete graph Kn for n>=3, and need to color r of...

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Accounting

If we have the complete graph Kn for n>=3, and need to color r of the vertices in Kn blue and the remaining n-r(=g) vertices purple. For any two vertices v, w in Kn color the edge {v,w} (1) blue if v,w are both blue (2) purple if v,w are both purple or (3) yellow if v,w have different colors. Assume that r>=g.

a) Show that for r=6 and g=3 (and n=9) the total number of blue and purple edges in K9 equals the number of yellow edges in K9

b) Show that the total number of blue and purple edges in Kn equals the number of yellow edges in Kn if and only if n=r+g, where g,r are consecutive triangular numbers. [The triangular numbers are defined recursively by t1 = 1, tn+1 = tn + (n+1), n>=1; so tn = n(n+1)/2. Hence t1 = 1, t2 = 3, t3 = 6,...].

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