Forbidden Subgraphs in Connected Graphs

TL;DR

This paper investigates the structure and enumeration of connected graphs avoiding certain subgraphs, deriving generating functions, asymptotic probabilities, and limiting distributions for sparse and near-sparse graphs, including multigraphs.

Contribution

It introduces differential recurrences for the generating functions of connected ree graphs and establishes their rational form in terms of Cayley's tree generating function, advancing enumeration and probabilistic understanding.

Findings

Derived differential recurrences for ree graphs' EGFs.

Proved EGFs are rational functions of the tree generating function.

Established asymptotic probabilities and distributions for sparse ree graphs.

Abstract

Given a set $\xi=\{H_1,H_2,...\}$ of connected non acyclic graphs, a $\xi$-free graph is one which does not contain any member of $% \xi$ as copy. Define the excess of a graph as the difference between its number of edges and its number of vertices. Let ${\gr{W}}_{k,\xi}$ be theexponential generating function (EGF for brief) of connected $\xi$-free graphs of excess equal to $k$ ($k \geq 1$). For each fixed $\xi$, a fundamental differential recurrence satisfied by the EGFs ${\gr{W}}_{k,\xi}$ is derived. We give methods on how to solve this nonlinear recurrence for the first few values of $k$ by means of graph surgery. We also show that for any finite collection $\xi$ of non-acyclic graphs, the EGFs ${\gr{W}}_{k,\xi}$ are always rational functions of the generating function, $T$, of Cayley's rooted (non-planar) labelled trees. From this, we prove that almost all connected graphs with $n$ nodes and $n+k$ edges are $\xi$-free, whenever $k=o(n^{1/3})$ and $|\xi| < \infty$ by means of Wright's inequalities and saddle point method. Limiting distributions are derived for sparse connected $\xi$-free components that are present when a random graph on $n$ nodes has approximately $\frac{n}{2}$ edges. In particular, the probability distribution that it consists of trees, unicyclic components, $...$, $(q+1)$-cyclic components all $\xi$-free is derived. Similar results are also obtained for multigraphs, which are graphs where self-loops and multiple-edges are allowed.