Comments on "Line Digraph Iterations and Connectivity Analysis of de Bruijn and Kautz Graphs"

The aim of this note is to present some counterexamples to the results in the paper by Du, Lyuu and Hsu published in this Transactions 1]. We present in this paper, for any d 2, a digraph G 2 L (d; 4) \ L (d ? 1; 4) such that LG 6 2 L (d ? 1; 5) and L 2 G 6 2 L (d; 6). Therefore, this is a counterexample to Lemma 3.2 and Theorem 3.3 of 1]. The digraphs we consider here are members of a family of bipartite digraphs, called BD(d; n), constructed by Fiol and Yebra 2]. See this paper for the deenition and some properties of these digraphs. The digraph BD(d; n) is d-regular and bipartite with partite sets V 0 = f0gZ n and V 1 = f1g Z n. One important fact about this family is that the line digraph LBD(d; n) is isomorphic to BD(d; dn). For any vertex (; i) of BD(d; d 2 + 1), we have that ? + 2 (; i) = V ? f(; i)g. Then, there is an unique path of length 2 between any pair of diierent vertices in V and there are no cycles of length 2 in BD(d; d 2 + 1). That is, all the vertices which appear in the tree formed by all paths of length at most 2 from x are diierent. If x and y are two diierent vertices of BD(d; d 2 +1) in the same partite set, x; y] will denote the unique path of length 2 from x to y. The vertex z such that xzy = x; y] will be written as bbx; y]. be vertices (not necessarily diierent) of B(d; d 2 + 1). We are going to prove that there exist d disjoint paths (one for each y i) of positive length at most 4 from x to the vertices y i. Since BD(d; d 2 +1) is vertex-symmetric, we can suppose that x 2 V 0. We consider rst all possible disjoint paths of length 1 from x to the vertices y j. Rearranging the subindices, we can suppose that these paths are P i = xy i , 1 i p, where 0 p d. After that, we take all possible disjoint paths of length 2 from x to the vertices y j that are also disjoint with the paths …