Completely Independent Spanning Trees on Some Interconnection Networks

Let T-1, T-2, ... , T-k 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 T-1, T-2, ... , T-k 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 Peterfalvi recently. In this note, we give a necessary condition for k-connected k-regular graphs with left perpendiculark/2right perpendicular 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.