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

Abstract

Self-dual $1$-configurations $(n_d{)}_1$ have the most $K_d$-separated Menger graph $\mathcal{Y}$ for connected self-dual configurations $(n_d)$. Such $\mathcal{Y}$ is most symmetric if it is $K_d$-ultrahomogeneous. In this work, such a graph $\mathcal{Y}$ 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 $K_4$. These are shown to share each copy of $K_3$ with two copies of $L(Q_3)$. Vertices and copies of $L(Q_3)$ in $\mathcal{Y}$ are the points and lines of a self-dual $({104}_{12}{)}_1$.