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

Abstract

The Steiner $k$-diameter $sdia{m}_k(G)$ of a graph $G$, introduced by Chartrand, Oellermann, Tian and Zou in 1989, is a natural generalization of the concept of classical diameter. When $k=2$, $sdia{m}_2(G)=diam(G)$ 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 is $\ell$ 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 is at most $\ell$.

23 pages, 5 figures. arXiv admin note: text overlap with arXiv:1703.03984, arXiv:1703.01410