Some remarks on sparsely connected isomorphism-free labeled graphs

Abstract

Given a set ζ = (H1, H2, · · ·) of connected non-acyclic graphs, a ζ-free graph is one which does not contain any member of ζ as induced subgraph. Our first purpose in this paper is to perform an investigation into the limiting distribution of labeled graphs and multigraphs (graphs with possible self-loops and multiple edges), with n vertices and approximately 1/2n edges, in which all sparse connected components are ζ-free. Next, we prove that for any finite collection ζ of multicyclic graphs almost all connected graphs with n vertices and n + o(n1/3) edges are ζ-free. The same result holds for multigraphs. © Springer-Verlag Berlin Heidelberg 2000.