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 \( (104_{12})_1 \).