Fault-tolerant designs in lattice networks on the Klein bottle

In this note, we consider triangular, square and hexagonal lattices on the flat Klein bottle, and find subgraphs with the property that for any $j$ vertices there exists a longest path (cycle) avoiding all of them. This completes work previously done in other lattices.

[1]  Tudor Zamfirescu,et al.  Highly non-concurrent longest cycles in lattice graphs , 2013, Discret. Math..

[2]  Werner Schmitz.  Über längste Wege und Kreise in Graphen , 1975 .

[3]  Tudor Zamfirescu,et al.  On longest paths and circuits in graphs. , 1976 .

[4]  Dharma P. Agrawal,et al.  Graph Theoretical Analysis and Design of Multistage Interconnection Networks , 1983, IEEE Transactions on Computers.

[5]  Hansjoachim Walther,et al.  Über die Nichtexistenz eines Knotenpunktes, durch den alle längsten Wege eines Graphen gehen , 1969 .

[6]  Tudor Zamfirescu,et al.  On Longest Paths in Triangular Lattice Graphs , 2010 .

[8]  H. Voss.  Cycles and Bridges in Graphs , 1991 .

[9]  Stanislav Jendrol',et al.  Exact Numbers of Longest Cycles with Empty Intersection , 1997, Eur. J. Comb..

[10]  Carol T. Zamfirescu,et al.  Intersecting longest paths and longest cycles: A survey , 2013, Electron. J. Graph Theory Appl..

[11]  Tudor Zamfirescu,et al.  A two-connected planar graph without concurrent longest paths , 1972 .

[12]  Hyeong-Seok Lim,et al.  Fault-Hamiltonicity of hypercube-like interconnection networks , 2005, 19th IEEE International Parallel and Distributed Processing Symposium.

[13]  Tudor Zamfirescu,et al.  Gallai’s property for graphs in lattices on the torus and the Möbius strip , 2016, Period. Math. Hung..

[14]  John P. Hayes,et al.  A Graph Model for Fault-Tolerant Computing Systems , 1976, IEEE Transactions on Computers.

[15]  Tudor Zamfirescu,et al.  Planar Lattice Graphs with Gallai’s Property , 2013, Graphs Comb..

[16]  Heinz-Jürgen Voss,et al.  Über Kreise in Graphen , 1974 .

[17]  Jung-Heum Park,et al.  Panconnectivity and pancyclicity of hypercube-like interconnection networks with faulty elements , 2007, Theor. Comput. Sci..