We considered Sn with n≥6 in Theorem 1 and Corollary 1. The situations of S4 and S5 are now discussed.
Suppose u and v are two vertices of S4. If |Fe|=n-3=1, then |
a1
D |=1 and |
a2
D |=|
a3
D |=0. The break dimension is a1 or a2. If a2 is the break dimension, dist(u, v)=1 or 2 or 3 according to Lemma 14. The following theorem can be obtained by exhaustively examining S4.
Theorem 2. Suppose u and v are two vertices of S4 with one edge fault, and a1a2a3 is a crucial permutation with respect to u and v. If a1 is the break dimension, S4 can embed a fault-free u-v path of length at least 4!-7 (4!-6) if dist(u, v) is odd (even). If a2 is the break dimension, S4 can embed a longest fault-free u-v
path if dist(u, v)=1 or 2, and a fault-free u-v path of length 4!-3 if dist(u, v)=3. The longest fault-free u-v path has length 4!-1 (4!-2) if dist(u, v)=1 (2).
If |Fe|=n-4=0, it is not difficult to embed a longest fault-free u-v path in S4. The discussion for S5 is similar, and the following two theorems can be obtained by exhaustively examining S5.
Theorem 3. Suppose u and v are two vertices of S5 with two edge faults, and a1a2a3a4 is a crucial permutation with respect to u and v. If |
a1
D |=2 and a1 is the break dimension, S5 can embed a fault-free u-v path of length at least 5!-5 (5!-8) if dist(u, u-v) is odd (eu-ven). If |
a1
D |=2 and a2 is the break dimension, or
|Da1 |=|
a2
D |=1 and a1 is the break dimension, S5 can embed a fault-free u-v path of length at least 5!-11 (5!-8) if dist(u, v) is odd (even). If |
a1
D |=|
a2
D |=1 and a3 is the break dimension, S5 can embed a longest fault-free u-v path if dist(u, v)=1 or 2, and a fault-free u-v path of length 5!-3 if dist(u, v)=3. The longest fault-free u-v path has length 5!-1 (5!-2) if dist(u, v)=1 (2).
Theorem 4. Suppose u and v are two vertices of S5 with one edge fault, and a1a2a3a4 is a crucial permutation with respect to u and v. If a1 is the break dimension, S5 can embed a longest fault-free u-v path whose length is 5!-1 (5!-2) if dist(u, v) is odd (even). If a2 is the break dimension, S5 can embed a fault-free u-v path of length at least 5!-3 (5!-4) if dist(u, v) is odd (even).
It is also noted that longest paths for fewer pairs of u and v can be determined in both S4 and S5 than Sn
with n≥6, because good fault-free P3’s for fewer pairs of u and v can be constructed in the former than the latter. In our method, a good fault-free P3 is prerequisite to a longest fault-free path.
The probabilities of the two exceptions in Theorem 1 can be analyzed as follows. Under the assumption that an edge fault has equal probability of falling into each dimension, we need to solve the
“partition of m” problem first, in order to compute the probabilities. The “partition of m” problem (see page 12 of [23]) asks the number of different ways to write a positive integer m as a sum of positive integers, disregarding their order. For example, m=4 can be written as 1+1+1+1 or 2+1+1 or 2+2 or 3+1 or 4. When m=3, 4, 5, 6, 7, 8, 9, and 10, the answers are 3, 5, 7, 11, 14, 20, 27, and 35, respectively. The
“partition of m” problem has been recognized as a quite difficult problem.
In order to estimate the probability of the exception (1), we assume that each dimension has equal probability of being the break dimension and dist(u, v) has equal probability of being any possible value.
Since dist(u, v)=1 or 2 or 3 by Lemma 14 and the break dimension may be a1 or an-2, the probability of the exception (1) for n=6 is computed as 31×21×31=181 . Similarly, the probabilities for n=7, 8, 9, 10, 11, 12,
and 13 are 301 , 421 , 661 , 841 , 1201 , 1621 , and 2101 , respectively. The probabilities tend to drop as n’s
increase. The probability of the exception (2) can be estimated similarly. Since the break dimension may be a1 or a2 or a3, the probability of the exception (2) is computed as
3 1 3
1× =19 as n=6, and 0 as n≠6.
It is still unknown whether the fault-free u-v paths we constructed for the two exceptions in Theorem 1 are the longest or not.
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1234
3214
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2134
3124
4231
3241
2341 3421
4321
3412
4312
1342
3142
4132 1432
2413
4213 1423
1243
4123
2143 α
α
δ χ
β
χ δ
β
Figure 1. The structure of S4.
2431
<****1>4
<****2>4
<****3>4
<****4>4
<****5>4
Figure 2. A P4 (shown in bolded lines) contained in S5.