Split graphs: combinatorial species and asymptotics

TL;DR

This paper studies the combinatorial structure and asymptotic enumeration of split graphs, extending previous results with species-theoretic methods and confirming that most split graphs are balanced.

Contribution

It provides species-theoretic generalizations of existing enumeration results for split graphs, including asymptotic counts and the proof that almost all split graphs are balanced.

Findings

Asymptotic enumeration of split graphs and related classes

Species-theoretic generalizations of previous results

Proof that almost all split graphs are balanced

Abstract

A split graph is a graph whose vertices can be partitioned into a clique and a stable set. We investigate the combinatorial species of split graphs, providing species-theoretic generalizations of enumerative results due to B\'ina and P\v{r}ibil (2015), Cheng, Collins, and Trenk (2016), and Collins and Trenk (2018). In both the labeled and unlabeled cases, we give asymptotic results on the number of split graphs, of unbalanced split graphs, and of bicolored graphs, including proving the conjecture of Cheng, Collins, and Trenk (2016) that almost all split graphs are balanced.