On the automorphism groups of edge-coloured digraphs (Q750453)

Let D be an edge-coloured digraph and C be the set of colours c with which at least one directed edge of D is coloured. For each \(x\in V(D)\) and each \(c\in C\), let \(\lambda_{in}(x;c)\) resp. \(\lambda_{out}(x;c))\) denote the number of directed edges with colour c and having x as head (resp. as tail). Define \(\lambda_{\max}(D)=\max \{\lambda_{\max}(D)+1,3\}.\) The author proves the following theorem which extends some classical results of R. Frucht and G. Sabidussi. Theorem: Let D be an edge-coloured weakly connected digraph of order n. Then for any integer \(k\geq \lambda (D)\), there exist infinitely many connected k-regular graphs G in which V(G) has a disjoint union \(\cup^{n}_{i=1}V_ i\) such that Aut G acts faithfully on the set \(\{V_ 1,V_ 2,...,V_ n\}\) by the natural action and the permutation group derived by its action is isomorphic to the colour-preserving automorphism group Aut D of D on V(D) as a permutation group. The lemmas 3, 4, 5 and 6 used to prove this theorem are given without proof (this is because the number of pages of each paper published in Proc. Japan Acad. usually does not exceed 5). Although the author says that the proofs of these lemmas are easy, but in fact the graphs given in these lemmas are not easy to construct. I have a copy of the original version of this paper and I find that the author uses very ingeneous ways to show that the graphs he constructed satisfy the requirements of the lemmas.