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.


Introduction
Fault-tolerance allows a system to continue function after a component failure without any interruption.It is one of the key criteria in deciding the structures of interconnection networks for parallel and distributed systems.A fault-tolerant system follows a pattern or design (along with a set of instructions), which is usually modeled as a graph (see e.g.[1], [5]), in which vertices and edges correspond to processing units and communication links, respectively.Here, we are presenting some fault-tolerant designs in the form of P j k -graphs or C j k -graphs, in which n processing units are interlinked, and l of these n units forming a chain or cycle of maximal length are used to solve some task.They can tolerate the failure of up to j components or communication links, keeping constant performance, due to which the designs presented here can be used to construct perfectly stable systems, while the concept of j-fault hamiltonicity (see e.g.[8], [9]) does not provide too stable systems, as the (maximal) length of the used circuit changes when failure occurs.

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Fault-tolerant designs in lattice networks on the Klein bottle | Ayesha Shabbir A finite P j k -graph (C j k -graph) is a k-connected graph such that for every choice of j vertices there exists a longest path (cycle) that misses the j chosen vertices.The smallest known P 1 1 -graph and planar P 1  1 -graph are shown in Fig 1(a) and (b), respectively.The first was found by H.-J. Voss and H. Walther [16] and T. Zamfirescu independently [18], while the second was presented by W. Schmitz in [10].See [18] and [19], for several small examples of P j k -and C j k -graphs (k = 1, 2, 3; j = 1, 2).In 1966, T. Gallai raised the question about the existence of P 1 1 -graphs, and in 1972, Zamfirescu refined it to P j k -and C j k -graphs [17].A similar question was also asked for planar graphs.The question raised in 2001, about the existence of such graphs in lattices (see [19]) also received a positive answer.See [2], [3], [7], [13] and [12], where the existence of P 1 k -and C j kgraphs (k = 1, 2; j = 1, 2) in all three (infinite) regular lattices, T (triangular), L (square) and H (hexagonal) is shown.Their orders are considerably larger.Comparatively smaller examples of P 1 k -and C j 2 -graphs for k = 1, 2 and j = 1, 2 in toroidal and Möbius strip lattices are given in [11] and [13].
To complete the previous investigation, in this note we consider (finite) triangular, square and hexagonal lattices on another interesting surface, the Klein bottle, and construct P 1 1 -, P 1 2 -, C 1 2 -and C 2 2 -graphs there.The graphs presented here are smaller than those obtained in other mentioned lattices (see the table given at the end).This was achieved in several cases not by embedding new graphs, but just by using the topology of the Klein bottle.The examples have not been found using computers; instead, modifications of known examples suitable for the new frame have been employed.
Note that the connectivity of any finite connected graph in T is at most 3, while in L or H it is at most 2.Moreover, it is worth mentioning that no P j 1 -or C j 2 -graph at all (whether planar or not) is known for j ≥ 3 (see [14], p. 79).
The following two lemmas are of great help.
Let G and H be graphs homeomorphic to the graphs G and H in Fig. 2 and Fig. 3(a), respectively.The graph H contains ten subgraphs isomorphic to the graph of Fig. 3(b).The variables v, x, y, z, t and w denote the number of vertices of degree 2 on paths corresponding to edges shown on the respective figures as well.

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Fault-tolerant designs in lattice networks on the Klein bottle | Ayesha Shabbir

Main Results
We start with the definition of lattices on the Klein bottle.
We consider an (m + 1) × (n + 1) parallelogram in T .Then by identifying opposite vertices on the boundary as indicated on Fig. 4(b), we obtain the T -lattice T K m,n on the Klein bottle.It has mn vertices.Similarly, we define the L-lattice L K m,n and the H-lattice H K m,n , according to Fig. 4(a) and (c), respectively.
We now state our results, which are distributed in three sections.In next section, P 1 1 -, C 1 2and C 2 2 -subgraphs of H-lattices are given.In the second and third sections, we consider Land T -lattices, respectively, and describe P 1 k -and C j 2 -subgraphs (k = 1, 2; j = 1, 2) of these lattices.
Proof.The existence of a planar C 2 2 -graph of order 600 in H K 30,32 is shown in Fig. 7.This graph is isomorphic to the graph H in Lemma 2 for y = z = 3, x = 8, t = 14 and w = 0.

L-lattices
As H ⊂ L, every subgraph of a H-lattice is a subgraph of some L-lattice, but not vice-versa.The graphs we present next are interesting because they are smaller than the corresponding ones obtained in H-lattices.
Proof.The graph shown in Fig. 8, is an embedding of Schmitz's graph of Fig. 1(b) into L K 6,4 .Since Schmitz's graph is conjectured to be the smallest planar P 1 1 -graph, it seems reasonable to say that our embedded graph and corresponding L-lattice might be of minimal order too.The last result of this section is as follows.

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Theorem 2.6.We have a planar C 2 2 -graph of order 315 in L K 26,18 .Proof.For y = 1, z = 2, x = 3, t = 9 and w = 0, the conditions of Lemma 2 are valid and the corresponding graph H is a planar C 2 2 -graph of order 315.Fig. 10 shows an embedding of H in L T 26,18 . www.ejgta.org

T -lattices
Since H ⊂ L ⊂ T , we may expect to get better results, that is graphs of smaller order, when embedding in T -lattices.
It is a conjecture that the graph of Fig. 1(a) is the smallest P 1 1 -graph [18].Hence, we may conjecture that our next result is best possible.Theorem 2.7.In T K 4,3 we have a spanning P 1 1 -graph.
Proof.A spanning P 1 1 -subgraph of T K 4,3 isomorphic to the graph of Fig. 1(a) is shown in Fig. 11.

Fault-tolerant designs
in lattice networks on the Klein bottle | Ayesha Shabbir

Theorem 2 . 8 .
There exists a planar P 1 2 -graph in T K 7,8 of order 48.Proof.The graph of Fig.12is a planar graph of order 48 in T K 7,8 .It is a P 1 2 -graph, being obtained as a particular case of Lemma 1, for y = 1, z = 0, and x = v = w = 2.

Theorem 2 . 9 .
There exists a planar C 2 2 -graph in T K 11,25 of order 235.Proof.The conditions of Lemma 2 are also verified by y = 1, z = 0, x = 2 and w = 1.Hence, the resulting graph H is a planar C 2 2 -graph of order 235.Fig. 13 reveals an embedding of it in T K 11,25 .Remark 2.1.The existence of spanning C 1 2 -graphs in L K 6,2 and T K 6,2 , is shown in Fig. 5(c) and (d), respectively.It is also follows from Theorem 1.