Further topics in connectivity

Continuing the study of connectivity, initiated in §4.1 of the Handbook, we survey here some (sufficient) conditions under which a graph or digraph has a given connectivity or edge-connectivity. First, we describe results concerning maximal (vertex- or edge-) connectivity. Next, we deal with conditions for having (usually lower) bounds for the connectivity parameters. Finally, some other general connectivity measures, such as one instance of the so-called “conditional connectivity,” are considered. For unexplained terminology concerning connectivity, see §4.1.Peer ReviewedPostprint (published version

Maximum Degree Growth of the Iterated Line Graph

Let # k denote the maximum degree of the k th iterated line graph L k (G). For any connected graph G that is not a path, the inequality # k+1 # 2# k - 2 holds. Niepel, Knor, and Soltes [3] have conjectured that there exists an integer K such that, for all k # K, equality holds; that is, the maximum degree # k attains the greatest possible growth. We prove this conjecture using induced subgraphs of maximum degree vertices and locally maximum vertices