Completely Independent Spanning Trees on Some Interconnection Networks

Kung-Jui PAI  Jinn-Shyong YANG  Sing-Chen YAO  Shyue-Ming TANG  Jou-Ming CHANG  

IEICE TRANSACTIONS on Information and Systems   Vol.E97-D    No.9    pp.2514-2517
Publication Date: 2014/09/01
Online ISSN: 1745-1361
DOI: 10.1587/transinf.2014EDL8079
Type of Manuscript: LETTER
Category: Information Network
completely independent spanning trees,  interconnection networks,  chordal rings,  

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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.