The distance-3 graph of the Biggs-Smith graph
نویسنده
چکیده
The distance-3 graph S3 of the Biggs-Smith graph S is shown to be: (a) a connected edge-disjoint union of 102 tetrahedra (copies of K4) and as such the K4-ultrahomogeneous Menger graph of a self-dual (1024)-configuration; (b) a union of 102 cuboctahedra, (copies of L(Q3)), with no 2 such cuboctahedra having a common chordless 4-cycle; (c) not a line graph. Moreover, S3 is shown to have a C-ultrahomogeneous property for C = {K4}∪ {L(Q3)} restricted to preserving a specific edge partition of L(Q3) into 2-paths, with each triangle (resp. each edge) shared by 2 copies of L(Q3) plus one of K4 (resp. 4 copies of L(Q4)). Both the distance-2 and distance-4 graphs, S 2 and S4, of 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 K4-ultrahomogeneous Menger graphs of self-dual (n4)-configurations that are edge-disjoint unions of several copies of K4, for positive integers n / ∈ {42, 102}.
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