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arXiv · 1904.02229

Existence of Regular Nut Graphs and the Fowler Construction

John Baptist Gauci·Tomaz Pisanski·Irene Sciriha
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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 $N(ρ)$ denote the set of integers $n$ such that there exists a regular nut graph of degree $ρ$ 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 $N(ρ)$ for $ρ> 4$ remains completely open.

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John Baptist Gauci, Tomaz Pisanski, Irene Sciriha. 2019-11-12. Existence of Regular Nut Graphs and the Fowler Construction. https://arxiv.org/abs/1904.02229

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