An Extremal Problem of Graphs with Diameter 2

281 If b is the term with no prime factor exceeding 3, there are also six possibilities. By section 4, every possibility requires S to include a term with a prime factor exceeding 11, which is forbidden. Thus S does not exist. une classe spéciale des diviseurs de la somme d'une série géometrique, C. Let 1 < k < p. We say that a graph has property P(p, k) if it has p points and every two of its points are joined by at least k paths of length : :~-: : 2. The aim of this note is to discuss the following problem. At least how many edges are in a graph with property P(p, k)? Denote this minimum by m(p, k). Construct a graph G,(p, k) with property P(p, k) as follows. Take two classes of points, k in the first class and p-k in the second, and take all the edges incident with at least one point in the first class. Thus Go(p, k) has j p)-(p-k) \2 2 edges. Murty [2] proved that if p ? 2(3 + V)k then m (p, k) = (2)-(p 2 k) and Go(p, k) is the only graph with property P(p, k) that has m (p, k) edges. He also suspected that the same result holds already for p > 2k. We shall show that this is not so, in fact p > 2(3 + V5)k is almost necessary for G,(p, k) to be an extremal graph, and we determine the asymptotic value of m ([ck ], k) for every constant I < c < 2(3 + V-5), where [x ] denotes the integer part of x. 7. 22 • 3 1 b-:~> c = P', d = 2q y => (c, d) satisfies [p, q, 2]. 8. b = 2 • 3y => a = p x (a, b) satisfies [p, 3, 2]. 9. b = 2 x, 3 }' a a = q y => (b, a) satisfies [2, q, I]. 10. b = 2x, 31 a > c = q y (b, c) satisfies [2, q, I]. 11. b = 3x, 2'11 a > a = 2q y (b a) satisfies [3, q, 2] .