The Existence of Completely Independent Spanning Trees for Some Compound Graphs

• Published in: IEEE transactions on parallel and distributed systems Vol. 31; no. 1; pp. 201 - 210
• Main Authors: Qin, Xiao-Wen, Hao, Rong-Xia, Chang, Jou-Ming
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.01.2020; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Apexes; Bipartite graph; Completely independent spanning tree; compound graph; Compounds; data center network architecture; Data centers; fault-tolerance; Graph theory; Graphs; Hypercubes; Network architecture; Routing; Servers; Trees (mathematics)
• ISSN: 1045-9219; 1558-2183
• DOI: 10.1109/TPDS.2019.2931904

Given two regular graphs G and H such that the vertex degree of G is equal to the number of vertices in H, the compound graph G(H) is constructed by replacing each vertex of G by a copy of Hand replacing each edge of G by an additional edge connecting random vertices in two corresponding copies of H, respectively, under the constraint that each vertex in G(H) is incident with only one additional edge, exactly. L-HSDC m is a compound graph G(H), where G is a hypercube Q m and H is a complete graph K m , which is defined by focusing on the connected relation between servers in the novel data center network HSDC m proposed in [30]. A set of k spanning trees in a graph G are called completely independent spanning trees (CISTs for short) if the paths joining every pair of vertices x and yin any two trees have neither vertex nor edge in common, except for x and y. In this paper, we give a sufficient condition for the existence of k CISTs in a kind of compound graph. Furthermore, a specific construction algorithm is provided. As corollaries of the main results, the existences of two CISTs form m ≥ 4; three CISTs form m ≥ 8 and four CISTs form m ≥ 10 in L-HSDC m (m) are gotten directly.