Split Graphs and Nordhaus-Gaddum Graphs

TL;DR

This paper characterizes Nordhaus-Gaddum graphs and split graphs using degree sequences, providing linear-time recognition algorithms, classifying their types, and exploring their interrelations to understand their growth patterns.

Contribution

It introduces degree sequence characterizations for NG-graphs and split graphs, establishes bijections between classes, and offers insights into their enumeration and growth behavior.

Findings

NG-graphs characterized by degree sequences

Linear-time recognition algorithm for NG-graphs

Bijections between NG-graph classes and split graphs

Abstract

A graph G is an NG-graph if \chi(G) + \chi(G complement) = |V(G)| + 1. We characterize NG-graphs solely from degree sequences leading to a linear-time recognition algorithm. We also explore the connections between NG-graphs and split graphs. There are three types of NG-graphs and split graphs can also be divided naturally into two categories, balanced and unbalanced. We characterize each of these five classes by degree sequence. We construct bijections between classes of NG-graphs and balanced and unbalanced split graphs which, together with the known formula for the number of split graphs on n vertices, allows us to compute the sizes of each of these classes. Finally, we provide a bijection between unbalanced split graphs on n vertices and split graphs on n-1 or fewer vertices providing evidence for our conjecture that the rapid growth in the number of split graphs comes from the balanced split graphs.