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The Steiner  k -diameterof a graph  G , introduced by Chartrand, Oellermann, Tian and Zou in 1989, is a natural generalization of the concept of classical diameter. When  k  = 2,is the classical diameter. The problem of determining the minimum size of a graph of order  n  whose diameter is at most  d  and whose maximum degree is ℓ was first introduced by Erdös and Rényi. In this paper, we generalize the above problem for Steiner  k -diameter, and study the problem of determining the minimum size of a graph of order  n  whose Steiner 3-diameter is at most  d  and whose maximum degree is at most ℓ.

Steiner 3-diameter, maximum degree and size of a graph