Mathematics – Combinatorics
Scientific paper
2010-02-02
Mathematics
Combinatorics
17 pages, 7 figures, 10 tables
Scientific paper
The distance-3 graph ${\mathcal S}^3$ of the Biggs-Smith graph $\mathcal S$ is shown to be: {\bf(a)} a connected edge-disjoint union of 102 tetrahedra (copies of $K_4$) and as such the $K_4$-ultrahomogeneous Menger graph of a self-dual $(102_4)$-configuration; {\bf(b)} a union of 102 cuboctahedra, (copies of $L(Q_3)$), with no two such cuboctahedra having a common chordless 4-cycle; {\bf(c)} not a line graph. Moreover, ${\mathcal S}^3$ is shown to have a $\mathcal C$-ultrahomogeneous property for ${\mathcal C}=\{K_4\}\cup\{L(Q_3)\}$ restricted to preserving a specific edge partition of $L(Q_3)$ into 2-paths, with each triangle (resp. each edge) shared by two copies of $L(Q_3)$ plus one of $K_4$ (resp. 4 copies of $L(Q_4)$). Both the distance-2 and distance-4 graphs, ${\mathcal S}^2$ and ${\mathcal S}^4$, of $\mathcal S$ appear in the context associated with the above mentioned edge partition. This takes us to ask whether there are any non-line-graphical connected $K_4$-ultrahomogeneous Menger graphs of self-dual $(n_4)$-configurations that are edge-disjoint unions of several copies of $K_4$, for positive integers $n\notin\{42,102\}$. Incidentally, two different Frucht diagrams of $\mathcal S$ allow to throw further light on its properties.
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