On a self-dual (1024)-configuration

A connected edge-disjoint union Y of 102 tetrahedra (copies ofK4) is constructed which is Menger graph of a self-dual (1024)-configuration and K4-ultrahomogeneous. As Y is not a line graph, we ask whether there exists a non-line-graphical K4-ultrahomogeneous Menger graph of a self-dual (n4)-configuration which is connected edge-disjoint union of n copies of K4, for n / ∈ {42, 102}. Moreover, Y is union of 102 cuboctahedra (copies of L(Q3)) with no two sharing a chordless 4-cycle and has an L(Q3)-ultrahomogeneity property restricted to preserving an edge partition of each L(Q3) into 2-paths, determined by the distance-i graphs Si of the Biggs-Smith graph S, for i = 1, 2, 3, 4. From this, it is concluded that Y = S3. In addition, Y has each edge (resp. triangle) shared exactly by 4 copies of L(Q3) (resp. two copies of L(Q3) plus one of K4). 1 K4-Ultrahomogeneous graphs Ultrahomogeneous (UH) graphs were treated in [5, 11, 13, 16, 17]. The present work deals with the following modified concept of UH (di)graph. Given a family C of (di)graphs closed under isomorphisms, a (di)graph G is said to be C-UH if every isomorphism between two induced members of C in G extends to an automorphism of G. If C is the isomorphism class of a (di)graph H , then G is said to be H-UH. In [15], C-UH graphs were defined and studied for a family C formed either by the complete graphs, or the disjoint unions of complete graphs, or the complements of those disjoint unions. A transformation of distance-transitive graphs into C-UH graphs that took in [9] from the Coxeter graph on 28 vertices onto the Klein graph on 56 vertices is applied in Section 4 below to the Biggs-Smith graph S [3, 4, 6] in order to yield a connected edge-disjoint union Y of 102 tetrahedra (copies of K4) which is K4-UH and a Menger graph [7] of a self-dual (1024)-configuration (Theorem 2). As Y is not a line graph and taking into account the main result of [8] and that the line graph of the d-cube is Kd-UH, for 3 ≤ d ∈ Z, we are able to pose the following: Question 1 Is there a non-line-graphical K4-UH Menger graph of a self-dual (n4)-configuration which is a connected edge-disjoint union of n copies of K4, for n / ∈ {42, 102}? In Section 2, definitions of restricted C-UH graphs are given towards more specific results of Sections 6–7 establishing finally that Y is the distance-3 graph of S.