Threshold functions for small subgraphs: an analytic approach

Abstract

We revisit the problem of counting the number of copies of a fixed graph in a random graph or multigraph, including the case of constrained degrees. Our approach relies heavily on analytic combinatorics and on the notion of patchwork to describe the possible overlapping of copies.This paper is a version, extended to include proofs, of the paper with the same title to be presented at the Eurocomb 2017 meeting.

Analytic combinatorics of connected graphs

“…• We apply an inclusion-exclusion technique to remove loops and double edges from multigraphs, turning them into graphs. Collet and coworkers [7] have recently extended this new approach to enumerate graphs with forbidden subgraphs, and to count the occurrences of subgraphs from a given family in random graphs. • New exact expressions for the generating functions of interesting families of (multi)graphs are derived, including multigraphs with a given excess and degree constraints (Proposition 1), and graphs and multigraphs without trees and unicycles (Propositions 3 and 6).…”

“…As already observed by Flajolet and coworkers [16] and Janson and coworkers [18], multigraphs are better suited for generating function manipulations than graphs. We use the model of Collet and coworkers [7], distinct but related to the one used by Flajolet and coworkers [16] and Janson and coworkers [18], and recall the link between the generating functions of graphs and multigraphs in Lemma 3. The difference between graphs and multigraphs is that multigraphs have labeled and oriented edges, and are permitted loops and multiple edges.…”

“…• We apply an inclusion-exclusion technique to remove loops and double edges from multigraphs, turning them into graphs. Collet et al (2017) have recently extended this new approach to enumerate graphs with forbidden subgraphs, and to count the occurrences of subgraphs from a given family in random graphs.…”

“…In Section 2.3, we improve their model to make it more compatible with the formalism of the symbolic method and of species theory . The generating functions of graphs with degree constraints were recently computed by de Panafieu and Ramos , and we apply and improve this result to enumerate graphs and multigraphs with minimum degree at least 2. We apply an inclusion‐exclusion technique to remove loops and double edges from multigraphs, turning them into graphs. Collet and coworkers have recently extended this new approach to enumerate graphs with forbidden subgraphs, and to count the occurrences of subgraphs from a given family in random graphs. New exact expressions for the generating functions of interesting families of (multi)graphs are derived, including multigraphs with a given excess and degree constraints (Proposition ), and graphs and multigraphs without trees and unicycles (Propositions and ).…”

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