Generalized recursive circulant graphs (GRCGs for short) are a generalization of recursive circulant graphs and provide a new type of topology for interconnection networks. A graph of n vertices is said to be s-pancyclic for some $3leqslant sleqslant n$ if it contains cycles of every length t for $sleqslant tleqslant n$. The pancyclicity of recursive circulant graphs was investigated by Araki and Shibata (Inf. Process. Lett. vol.81, no.4, pp.187-190, 2002). In this paper, we are concerned with the s-pancyclicity of GRCGs.

Let T1,T2,...,Tk be spanning trees in a graph G. If, for any two vertices u,v of G, the paths joining u and v on the k trees are mutually vertex-disjoint, then T1,T2,...,Tk are called completely independent spanning trees (CISTs for short) of G. The construction of CISTs can be applied in fault-tolerant broadcasting and secure message distribution on interconnection networks. Hasunuma (2001) first introduced the concept of CISTs and conjectured that there are k CISTs in any 2k-connected graph. Unfortunately, this conjecture was disproved by Péterfalvi recently. In this note, we give a necessary condition for k-connected k-regular graphs with ⌊k/2⌋ CISTs. Based on this condition, we provide more counterexamples for Hasunuma's conjecture. By contrast, we show that there are two CISTs in 4-regular chordal rings CR(N,d) with N=k(d-1)+j under the condition that k ≥ 4 is even and 0 ≤ j ≤ 4. In particular, the diameter of each constructed CIST is derived.