The Existence of Completely Independent Spanning Trees for Some Compound Graphs

Given two regular graphs <inline-formula><tex-math notation="LaTeX">$G$</tex-math><alternatives><mml:math><mml:mi>G</mml:mi></mml:math><inline-graphic xlink:href="hao-ieq1-2931904.gif"/></alternatives></inline-formula> and <inline-formula><tex-math notation="LaTeX">$H$</tex-math><alternatives><mml:math><mml:mi>H</mml:mi></mml:math><inline-graphic xlink:href="hao-ieq2-2931904.gif"/></alternatives></inline-formula> such that the vertex degree of <inline-formula><tex-math notation="LaTeX">$G$</tex-math><alternatives><mml:math><mml:mi>G</mml:mi></mml:math><inline-graphic xlink:href="hao-ieq3-2931904.gif"/></alternatives></inline-formula> is equal to the number of vertices in <inline-formula><tex-math notation="LaTeX">$H$</tex-math><alternatives><mml:math><mml:mi>H</mml:mi></mml:math><inline-graphic xlink:href="hao-ieq4-2931904.gif"/></alternatives></inline-formula>, the <italic>compound graph</italic> <inline-formula><tex-math notation="LaTeX">$G(H)$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>G</mml:mi><mml:mo>(</mml:mo><mml:mi>H</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="hao-ieq5-2931904.gif"/></alternatives></inline-formula> is constructed by replacing each vertex of <inline-formula><tex-math notation="LaTeX">$G$</tex-math><alternatives><mml:math><mml:mi>G</mml:mi></mml:math><inline-graphic xlink:href="hao-ieq6-2931904.gif"/></alternatives></inline-formula> by a copy of <inline-formula><tex-math notation="LaTeX">$H$</tex-math><alternatives><mml:math><mml:mi>H</mml:mi></mml:math><inline-graphic xlink:href="hao-ieq7-2931904.gif"/></alternatives></inline-formula> and replacing each edge of <inline-formula><tex-math notation="LaTeX">$G$</tex-math><alternatives><mml:math><mml:mi>G</mml:mi></mml:math><inline-graphic xlink:href="hao-ieq8-2931904.gif"/></alternatives></inline-formula> by an additional edge connecting random vertices in two corresponding copies of <inline-formula><tex-math notation="LaTeX">$H$</tex-math><alternatives><mml:math><mml:mi>H</mml:mi></mml:math><inline-graphic xlink:href="hao-ieq9-2931904.gif"/></alternatives></inline-formula>, respectively, under the constraint that each vertex in <inline-formula><tex-math notation="LaTeX">$G(H)$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>G</mml:mi><mml:mo>(</mml:mo><mml:mi>H</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="hao-ieq10-2931904.gif"/></alternatives></inline-formula> is incident with only one additional edge, exactly. <inline-formula><tex-math notation="LaTeX">$L$</tex-math><alternatives><mml:math><mml:mi>L</mml:mi></mml:math><inline-graphic xlink:href="hao-ieq11-2931904.gif"/></alternatives></inline-formula>-<inline-formula><tex-math notation="LaTeX">$HSDC_m(m)$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>H</mml:mi><mml:mi>S</mml:mi><mml:mi>D</mml:mi><mml:msub><mml:mi>C</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>m</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math><inline-graphic xlink:href="hao-ieq12-2931904.gif"/></alternatives></inline-formula> is a compound graph <inline-formula><tex-math notation="LaTeX">$G(H)$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>G</mml:mi><mml:mo>(</mml:mo><mml:mi>H</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="hao-ieq13-2931904.gif"/></alternatives></inline-formula>, where <inline-formula><tex-math notation="LaTeX">$G$</tex-math><alternatives><mml:math><mml:mi>G</mml:mi></mml:math><inline-graphic xlink:href="hao-ieq14-2931904.gif"/></alternatives></inline-formula> is a hypercube <inline-formula><tex-math notation="LaTeX">$Q_m$</tex-math><alternatives><mml:math><mml:msub><mml:mi>Q</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:math><inline-graphic xlink:href="hao-ieq15-2931904.gif"/></alternatives></inline-formula> and <inline-formula><tex-math notation="LaTeX">$H$</tex-math><alternatives><mml:math><mml:mi>H</mml:mi></mml:math><inline-graphic xlink:href="hao-ieq16-2931904.gif"/></alternatives></inline-formula> is a complete graph <inline-formula><tex-math notation="LaTeX">$K_m$</tex-math><alternatives><mml:math><mml:msub><mml:mi>K</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:math><inline-graphic xlink:href="hao-ieq17-2931904.gif"/></alternatives></inline-formula>, which is defined by focusing on the connected relation between servers in the novel data center network <inline-formula><tex-math notation="LaTeX">$HSDC_m(m)$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>H</mml:mi><mml:mi>S</mml:mi><mml:mi>D</mml:mi><mml:msub><mml:mi>C</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>m</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math><inline-graphic xlink:href="hao-ieq18-2931904.gif"/></alternatives></inline-formula> proposed in [30]. A set of <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="hao-ieq19-2931904.gif"/></alternatives></inline-formula> spanning trees in a graph <inline-formula><tex-math notation="LaTeX">$G$</tex-math><alternatives><mml:math><mml:mi>G</mml:mi></mml:math><inline-graphic xlink:href="hao-ieq20-2931904.gif"/></alternatives></inline-formula> are called <italic>completely independent spanning trees</italic> (CISTs for short) if the paths joining every pair of vertices <inline-formula><tex-math notation="LaTeX">$x$</tex-math><alternatives><mml:math><mml:mi>x</mml:mi></mml:math><inline-graphic xlink:href="hao-ieq21-2931904.gif"/></alternatives></inline-formula> and <inline-formula><tex-math notation="LaTeX">$y$</tex-math><alternatives><mml:math><mml:mi>y</mml:mi></mml:math><inline-graphic xlink:href="hao-ieq22-2931904.gif"/></alternatives></inline-formula> in any two trees have neither vertex nor edge in common, except for <inline-formula><tex-math notation="LaTeX">$x$</tex-math><alternatives><mml:math><mml:mi>x</mml:mi></mml:math><inline-graphic xlink:href="hao-ieq23-2931904.gif"/></alternatives></inline-formula> and <inline-formula><tex-math notation="LaTeX">$y$</tex-math><alternatives><mml:math><mml:mi>y</mml:mi></mml:math><inline-graphic xlink:href="hao-ieq24-2931904.gif"/></alternatives></inline-formula>. In this paper, we give a sufficient condition for the existence of <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="hao-ieq25-2931904.gif"/></alternatives></inline-formula> 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 for <inline-formula><tex-math notation="LaTeX">$m\geq 4$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>m</mml:mi><mml:mo>≥</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="hao-ieq26-2931904.gif"/></alternatives></inline-formula>; three CISTs for <inline-formula><tex-math notation="LaTeX">$m\geq 8$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>m</mml:mi><mml:mo>≥</mml:mo><mml:mn>8</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="hao-ieq27-2931904.gif"/></alternatives></inline-formula> and four CISTs for <inline-formula><tex-math notation="LaTeX">$m\geq 10$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>m</mml:mi><mml:mo>≥</mml:mo><mml:mn>10</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="hao-ieq28-2931904.gif"/></alternatives></inline-formula> in <inline-formula><tex-math notation="LaTeX">$L$</tex-math><alternatives><mml:math><mml:mi>L</mml:mi></mml:math><inline-graphic xlink:href="hao-ieq29-2931904.gif"/></alternatives></inline-formula>-<inline-formula><tex-math notation="LaTeX">$HSDC_m(m)$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>H</mml:mi><mml:mi>S</mml:mi><mml:mi>D</mml:mi><mml:msub><mml:mi>C</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>m</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math><inline-graphic xlink:href="hao-ieq30-2931904.gif"/></alternatives></inline-formula> are gotten directly.

[1]  Toru Hasunuma,et al.  Completely independent spanning trees in the underlying graph of a line digraph , 2000, Discret. Math..

[2]  Kung-Jui Pai,et al.  Dual-CISTs: Configuring a Protection Routing on Some Cayley Networks , 2019, IEEE/ACM Transactions on Networking.

[3]  Yoshihide Igarashi,et al.  Reliable broadcasting and secure distributing in channel networks , 1997, Proceedings of the 1997 International Symposium on Parallel Architectures, Algorithms and Networks (I-SPAN'97).

[4]  Haitao Wu,et al.  BCube: a high performance, server-centric network architecture for modular data centers , 2009, SIGCOMM '09.

[5]  Jou-Ming Chang,et al.  Completely Independent Spanning Trees on Complete Graphs, Complete Bipartite Graphs and Complete Tripartite Graphs , 2013 .

[6]  Yota Otachi,et al.  Completely Independent Spanning Trees in (Partial) k-Trees , 2015, Discuss. Math. Graph Theory.

[7]  Toru Hasunuma,et al.  Completely independent spanning trees in torus networks , 2012, Networks.

[8]  Jianxi Fan,et al.  Constructing completely independent spanning trees in crossed cubes , 2017, Discret. Appl. Math..

[9]  Alon Itai,et al.  The Multi-Tree Approach to Reliability in Distributed Networks , 1988, Inf. Comput..

[10]  Amin Vahdat,et al.  A scalable, commodity data center network architecture , 2008, SIGCOMM '08.

[11]  Toru Hasunuma,et al.  Completely Independent Spanning Trees in Maximal Planar Graphs , 2002, WG.

[12]  Qinghai Liu,et al.  Degree condition for completely independent spanning trees , 2016, Inf. Process. Lett..

[13]  Hung-Yi Chang,et al.  A Note on the Degree Condition of Completely Independent Spanning Trees , 2015, IEICE Trans. Fundam. Electron. Commun. Comput. Sci..

[14]  Olivier Togni,et al.  Completely independent spanning trees in some regular graphs , 2014, Discret. Appl. Math..

[15]  Myung M. Bae,et al.  Edge Disjoint Hamiltonian Cycles in k-Ary n-Cubes and Hypercubes , 2003, IEEE Trans. Computers.

[16]  János Tapolcai,et al.  Sufficient conditions for protection routing in IP networks , 2013, Optim. Lett..

[17]  Haitao Wu,et al.  Scalable and Cost-Effective Interconnection of Data-Center Servers Using Dual Server Ports , 2011, IEEE/ACM Transactions on Networking.

[18]  Toru Araki,et al.  Dirac's Condition for Completely Independent Spanning Trees , 2014, J. Graph Theory.

[19]  Lei Shi,et al.  Dcell: a scalable and fault-tolerant network structure for data centers , 2008, SIGCOMM '08.

[20]  Cheng-Kuan Lin,et al.  Toward the completely independent spanning trees problem on BCube , 2017, 2017 IEEE 9th International Conference on Communication Software and Networks (ICCSN).

[21]  Yu-Chee Tseng,et al.  Efficient Broadcasting in Wormhole-Routed Multicomputers: A Network-Partitioning Approach , 1999, IEEE Trans. Parallel Distributed Syst..

[22]  Sun-Yuan Hsieh,et al.  Constructing edge-disjoint spanning trees in locally twisted cubes , 2009, Theor. Comput. Sci..

[23]  Toru Hasunuma,et al.  Minimum Degree Conditions and Optimal Graphs for Completely Independent Spanning Trees , 2015, IWOCA.

[24]  Kung-Jui Pai,et al.  Constructing two completely independent spanning trees in hypercube-variant networks , 2016, Theor. Comput. Sci..

[25]  Kung-Jui Pai,et al.  Completely Independent Spanning Trees on Some Interconnection Networks , 2014, IEICE Trans. Inf. Syst..

[26]  Qinghai Liu,et al.  Ore's condition for completely independent spanning trees , 2014, Discret. Appl. Math..

[27]  Kung-Jui Pai,et al.  Parallel Construction of Independent Spanning Trees on Enhanced Hypercubes , 2015, IEEE Transactions on Parallel and Distributed Systems.