arXiv:1002.0588v16 [math.CO] 27 Feb 2011 On the Biggs-Smith graph and its powers Italo J. Dejter University of Puerto Rico Rio Piedras, PR 00936-8377 ijdejter@uprrp.edu Abstract A method of construction of C-ultrahomogeneous (or C-UH) graphs is applied to the Biggs-Smith graph S , seen in the process as a {C9}P 4 -UH graph, to show that S 3 is a connected edge-disjoint union of 102 tetrahedra (or K4 s) as well as a union of 102 cuboctahedra (or L(Q3) s) with no common 4-holes, and that it has a C-UH property (C = {K4}∪{L(Q3)}) restricted to preserving a specific edge decomposition of L(Q3) into 2- paths, with each triangle (resp. edge) shared exactly by two L(Q3) s and one K4 (resp. four L(Q4) s). This S 3 is the Menger graph of a self-dual (1024 )-configuration. Moreover, both S 2 and S 4 appear in the context associated to the above mentioned edge decomposition. 1 Introduction A method of construction of C -ultrahomogeneous (or C -UH) graphs from cubic distance-transitive graphs [9] that allowed for example to transform the Coxeter graph on 28 vertices into the Klein graph on 56 vertices [10] is applied to the Biggs-Smith graph S [2, 4] seen in the process to be a {C 9 } P4 -UH graph in order to show that the cube Y = S 3 of S is a connected edge-disjoint union of 102 tetrahedra (or K 4 s) as well as a union of 102 cuboctahedra (or L(Q 3 ) s) without common 4-holes (i.e. chordless 4-cycles), possessing a C -UH property for C = {K 4 }∪{L(Q 3 )} restricted to preserving a specific edge decomposition of L(Q 3 ) into 2-paths and having each triangle (resp. edge) of Y shared exactly by two L(Q 3 ) s and one K 4 (resp. four L(Q 3 ) s). This Y is the Menger graph [6] of a self-dual (102 4 )-configuration, but not a line graph. (This contrasts with the (42 4 )-configuration of [7] whose non-line-graphical Menger graph X is {K 4 ,K 2,2,2 } K2 -UH, namely the edge-disjoint union of 42 tetrahedra as well as the edge-disjoint union X of 21 octahedra (K 2,2,2 ) with the C -UH property for C = {K 4 }∪{K 2,2,2 } and each edge (or copy of K 2 ) shared tightly by one K 4 and one K 2,2,2 .) Through the paper, we will need from [2, 4] that S is distance-transitive, hamil- tonian, cubic, with order n = 102, diameter d = 7, girth g = 9, arc-transitivity 1