On a $K_4$-UH self-dual 1-configuration $(102_4)_1$

TL;DR

This paper constructs a highly symmetric self-dual 1-configuration with 102 points and explores its relation to cuboctahedral graphs and complete graphs, revealing specific sharing properties of triangles.

Contribution

It presents a new self-dual 1-configuration with a K_4-ultrahomogeneous Menger graph and details its relation to cuboctahedral graphs and complete graphs.

Findings

Constructed a self-dual 1-configuration with 102 points.

Established the relation between cuboctahedral graphs and complete graphs.

Showed each triangle is shared by exactly two cuboctahedral graphs.

Abstract

Self-dual 1-configurations $(n_d)_1$ possess their Menger graph $\mathcal Y$ most $K_4$-separated among connected self-dual configurations $(n_d)$. Such $\mathcal Y$ is most symmetric if $K_d$-ultrahomogeneous. In this work, such a $\mathcal Y$ is presented for $(n,d)=(102,4)$ and shown to relate $n$ copies of the cuboctahedral graph $L(Q_3)$ to the $n$ copies of $K_d$; these are shown to share each copy of $K_3$ exactly with two copies of $L(Q_3)$.