Existence of Regular Nut Graphs and the Fowler Construction

TL;DR

This paper investigates the existence of regular nut graphs, introduces a generalized Fowler construction for generating higher-order nut graphs, and characterizes the sets of possible orders for degrees 3 and 4.

Contribution

It provides a complete characterization of regular nut graphs for degrees 3 and 4 and introduces a new local enlargement method for constructing nut graphs.

Findings

N(3) = {12} ∪ {2k : k ≥ 9}

N(4) = {8,10,12} ∪ {n : n ≥ 14}

Open problem for degrees greater than 4

Abstract

In this paper the problem of the existence of regular nut graphs is addressed. A generalization of Fowler's Construction which is a local enlargement applied to a vertex in a graph is introduced to generate nut graphs of higher order. Let $N(\rho)$ denote the set of integers $n$ such that there exists a regular nut graph of degree $\rho$ and order $n$. It is proven that $N(3) = \{12\} \cup \{2k : k \geq 9\}$ and that $N(4) = \{8,10,12\} \cup \{n: n \geq 14\}$. The problem of determining $N(\rho)$ for $\rho > 4$ remains completely open.