On {K4, K2, 2, 2} ultrahomogeneity

The existence of a connected 12-regular {K4,K2,2,2}-ultrahomogeneous graph G is established, (i.e. each isomorphism between two copies of K4 or K2,2,2 in G extends to an automorphism of G), with the 42 ordered lines of the Fano plane as vertices and an adjacency resembling that of star Cayley graphs. The graph G can be expressed in a unique way both as the edge-disjoint union of 42 induced copies of K4 and as the edgedisjoint union of 21 induced copies of K2,2,2, with no more copies of K4 or K2,2,2 existing in G. Moreover, each edge of G is shared by exactly one copy of K4 and one of K2,2,2. While the line graphs of d-cubes, (3 ≤ d ∈ ZZ), are {Kd,K2,2}-ultrahomogeneous, G is not even line-graphical. In addition, self-dual configurations associated to G with 2-arc-transitive, arc-transitive and semisymmetric Levi graphs are considered, as well as relevant toroidal subgraphs of G, comprising 21 quadruple-star-of-David graphs and 7 star Cayley graphs.