On C-ultrahomogeneous graphs and digraphs

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 K3fastened {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 digraph 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.