Consider the complete graph Kn for ft ≥ 3. Color r of the vertices in Kn red and the remaining n - r ( = g) vertices green. For any two vertices v, w in Kn color the edge {u, w} (1) red if v, w are both red; (2) green if v, w are both green; or (3) blue 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 red and green edges in K9 equals the number of blue edges in K9.
(b) Show that the total number of red and green edges in Kn equals the number of blue 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 + l)/2. Hence
t1 = 1, t2 =3, t3 = 6, . .. .]