Limit law for number of components of fixed sizes of graphs with degree one or two

TL;DR

This paper establishes that in graphs with vertices of degree 1 or 2, the counts of components of fixed sizes follow a normal distribution asymptotically, extending the result to multigraphs.

Contribution

It proves a limit law for the distribution of component sizes in degree-restricted graphs, including multigraphs, which was previously unestablished.

Findings

Component counts of fixed sizes are asymptotically normally distributed.

The result applies to both simple graphs and multigraphs.

The proof extends classical limit theorems to degree-restricted graph classes.

Abstract

We consider graphs with vertices of degree 1 or 2 and prove that the numbers of components of sizes 2 to q have a limit normal distribution for any q > 1. The result is also extended to multigraphs.