On the number of biased graphs

TL;DR

This paper investigates the enumeration of biased graphs, showing that most simple biased graphs on n vertices are essentially constructed from complete graphs with certain cycle subsets, on a logarithmic scale.

Contribution

It establishes that the total number of simple biased graphs on n vertices is asymptotically bounded by those constructed from complete graphs with arbitrary Hamilton cycle subsets.

Findings

Most simple biased graphs are constructed from complete graphs with cycle subsets.

The total number of biased graphs does not asymptotically exceed elementary constructions.

The enumeration is characterized on a logarithmic scale.

Abstract

A biased graph is a graph $G$, together with a distinguished subset $\mathcal{B}$ of its cycles so that no Theta-subgraph of $G$ contains precisely two cycles in $\mathcal{B}$. A large number of biased graphs can be constructed by choosing $G$ to be a complete graph, and $\mathcal{B}$ to be an arbitrary subset of its Hamilton cycles. We show that, on the logarithmic scale, the total number of simple biased graphs on $n$ vertices does not asymptotically exceed the number that can be constructed in this elementary way.