arXiv:0904.1096v1 [math.CO] 7 Apr 2009 On C -ultrahomogeneous graphs and digraphs Italo J. Dejter University of Puerto Rico Rio Piedras, PR 00931-3355 idejter@uprrp.edu Abstract The notion of a C-ultrahomogeneous graph, due to Isaksen et al., is adapted for digraphs and studied for the twelve cubic distance transitive graphs, with C formed by g-cycles and (k - 1)-paths, where g = girth and k = arc-transitivity. Excluding the Petersen, Heawood and Foster (90 vertices) graphs, one can go further by considering the (k - 1)-powers of g-cycles under orientation assignments provided by the initial study. This allows the construction of fastened C-ultrahomogeneous graphs with C formed by copies of K3, K4, C7 and L(Q3), for the Pappus, Desargues, Coxeter and Biggs-Smith graphs. In particular, the Biggs-Smith graph yields a connected edge-disjoint union of 102 copies of K4 which is a non-line-graphical Menger graph of a self-dual (1024)-configuration, a K3- fastened {K4,L(Q3)}-ultrahomogeneous graph. This contrasts with the self-dual (424)-configuration of [5], whose non-line-graphical Menger graph is K2-fastened {K4,K2,2,2}-ultrahomogeneous. Among other results, a strongly connected  C4-ultrahomogeneous di- graph on 168 vertices and 126 pairwise arc-disjoint 4-cycles is obtained, with regular indegree and outdegree 3 and no circuits of lengths 2 and 3, by altering a definition of the Coxeter graph via pencils of ordered lines of the Fano plane in which pencils are replaced by ordered pencils. 1 Introduction Given a collection C of (di)graphs closed under isomorphisms, a (di)graph G is C -ultrahomogeneous (or C -UH) if every isomorphism between two G-induced members of C extends to an automorphism of G. If C = {H } is the isomorphism class of a graph H , we say that such a G is H -UH. In [8], C -UH graphs are defined and studied when C is the collection of (a) complete graphs, or (b) disjoint unions of complete graphs, or (c) complements of those unions. In [5], a {K 4 ,K 2,2,2 }-UH is given that fastens objects of (a) and (c), namely K 4 and K 2,2,2 . We may consider a graph G as a digraph by considering each edge of G as a pair of oppositely oriented (or OO) arcs. Let M be a sub(di)graph of a (di)graph H and let G be both an M -UH and H -UH (di)graph. We say that G is a fastened (H, M )-UH (di)graph if given a 1