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Pendant appearances and components in random graphs from structured classes

Colin McDiarmid

Abstract

We consider random graphs sampled uniformly from a structured class of graphs, such as the class of graphs embeddable in a given surface. We sharpen and extend earlier results on pendant appearances, concerning for example numbers of leaves; and obtain results on the asymptotic distribution of components other than the giant component, under quite general conditions.

Pendant appearances and components in random graphs from structured classes

Abstract

We consider random graphs sampled uniformly from a structured class of graphs, such as the class of graphs embeddable in a given surface. We sharpen and extend earlier results on pendant appearances, concerning for example numbers of leaves; and obtain results on the asymptotic distribution of components other than the giant component, under quite general conditions.

Paper Structure

This paper contains 7 sections, 13 theorems, 103 equations.

Table of Contents

  1. Introduction and statement of results
  2. Pendant appearances in the random graph $R_n$
  3. Components of the random graph $R_n$
  4. Related work
  5. Proofs for results on pendant appearances
  6. Proofs for results on components
  7. Concluding Remarks

Key Result

Theorem \oldthetheorem

Let the class $\mathcal{A}$ of graphs satisfy $0< \rho_\mathcal{A} < \infty$. Let $H^{\bullet}$ be a vertex-rooted connected graph, let $\alpha = \rho_{\!\mathcal{A}}^{\; v(H)}/{\rm aut\,}\,H^{\bullet}$, and let $0<\varepsilon<1$. Then there exists $\nu>0$ depending on $\rho_\mathcal{A}, H^{\bullet}

Theorems & Definitions (25)

  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • Corollary \oldthetheorem
  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • Corollary \oldthetheorem
  • Corollary \oldthetheorem
  • proof : Proof of Theorem \ref{['thm.rhoH']} (a)
  • proof : Proof of Theorem \ref{['thm.rhoH']} (b)
  • proof : Proof of Corollary \ref{['cor.rhoHnoroot']} from Theorem \ref{['thm.rhoH']}
  • ...and 15 more