On the Diameter of the Generalized Graphs UG(B)(n, m), n(2) < m <= n(3)

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(1)On the Diameter of the Generalized Undirected de Bruijn Graphs UGB(n, m), n 2 < m ≤ n 3. Jyhmin Kuo and Hung-Lin Fu Department of Applied Mathematics, National Chiao Tung University, Hsin Chu, Taiwan 30050, Republic of China. The generalized de Bruijn digraph GB (n, m) is the digraph (V , A) where V = {0, 1, . . . , m − 1} and (i , j ) ∈ A if and only if j ≡ in + α (mod m) for some α ∈ {0, 1, 2, . . . , n − 1}. By replacing each arc of GB (n, m) with an undirected edge and eliminating loops and multi-edges, we obtain the generalized undirected de Bruijn graph UGB (n, m). In this article, we prove that when 2n 2 ≤ m ≤ n 3 the diameter of UGB (n, m) is equal to 3. We also show that for pairs (n, m) where n 2 < m < 2n 2 the diameter of UGB (n, m) can be 2 or 3. © 2008 Wiley Periodicals, Inc. NETWORKS, Vol. 52(4), 180–182 2008. Keywords: generalized de Bruijn graph; diameter; undirected graph. 1. INTRODUCTION All graphs considered in this study are undirected, loopless, and without multi-edges. For graph theory terminology, we follow [9]. For brevity, [a, b] = {a, a+1, . . . , b} is defined here for non-negative integers a and b where a < b. Imase and Itoh [6] were the first to generalize the wellknown de Bruijn network B(d, n), independently followed by Reddy et al. [8]. The generalized de Bruijn digraph GB (n, m) is the directed graph, whose vertices are 0, 1, . . . , m − 1, and whose directed edges (arcs) are of the form i → in + α (mod m),. ∀i ∈ [0, m − 1] and ∀α ∈ [0, n − 1].. The generalized undirected de Bruijn graph is the undirected graph derived from the generalized de Bruijn digraph by replacing directed edges with undirected edges and omitting loops and multi-edges. Such a graph is denoted here by. Received June 2006; accepted September 2007 Correspondence to: J. Kuo, No. 37, Minxiang 1st St., East District, Hsinchu City 300, Taiwan; e-mail: jyhminkuo@gmail.com DOI 10.1002/net.20228 Published online 11 January 2008 in Wiley InterScience (www. interscience.wiley.com). © 2008 Wiley Periodicals, Inc.. NETWORKS—2008—DOI 10.1002/net. UGB (n, m). The set of neighbors of any vertex i in UGB (n, m) is N(i) = R(i) ∪ L(i), where R(i) = {in + α. (mod m) : α ∈ [0, n − 1]}. L(i) = { j : jn + β ≡ i. and. (mod m),. where β ∈ [0, n − 1], j ∈ [0, m − 1]}. Therefore, if j ∈ R(i) then i ∈ L(j). Imase and Itoh [6] proved that the generalized de Bruijn digraph GB (n, m) is (n − 1)-connected and its diameter is bounded above by logn m. Therefore, UGB (n, m) possesses the same properties. In the study of fault tolerance and transmission delay of networks, the connectivity and diameter of the graph are two very important parameters; these have been thoroughly studied by many authors [3, 4, 6, 10]. Since the de Bruijn graphs B(d, n) are known to have short diameters and simple routing strategies, they have been widely used as models for communication networks and multiprocessor systems [4]. However, one of the disadvantages of de Bruijn graphs B(d, n) is the restriction on the number of vertices d n . The generalized de Bruijn graphs retain all of the properties of the de Bruijn graphs, but have no restrictions on the number of vertices [4]. So, determining the connectivity and diameter of UGB (n, m) is of relevant interest and importance. Caro et al. [2] proved that UGB (n, n(n + 1)) has a w-widediameter of 5 for w = 2(n − 1). Escuadro and Muga [5] proved that UGB (n, n2 ) is 2(n − 1)-regular and has diameter 2; in addition, they showed that the w-wide-diameter of UGB (n, n2 ) is 4 for w = 2(n − 1) and n ≥ 4. Nochefranca and Sy [7] proved that the diameter of UGB (n, n(n2 + 1)) is 4 for odd n ≥ 3. Caro and Zeratsion [3] recently proved that the diameter of UGB (n, m) is 2 for m ∈ [n + 1, n2 ], and 3 for m ∈ [n2 + 1, n3 ] where n divides m. Caro et al. [1] also provided an upper bound for the diameter of UGB (n, n2 + 1) when n ≥ 5 is odd. This work shows that the diameter of UGB (n, m) is 3 whenever n ≥ 2 and 2n2 ≤ m ≤ n3 . Notably, for n2 < m < 2n2 , there are pairs (n, m) such that the diameter of UGB (n, m) is 2 or 3. This work also verifies that the diameter of UGB (n, n2 +1) is 3 and the diameter of UGB (n, n2 +2) is 2..

(2) 2. UGB (n, m), 2n 2 ≤ m ≤ n 3 Let dG (x, y) denote the distance between two vertices x and y in a graph (or directed graph) G, and let d(G) denote the diameter of the graph G. We use < u, . . . , v > to denote a path from u to v in G. Imase and Itoh [6] proved that the diameter of the generalized de Bruijn digraph GB (n, m) is bounded above by logn m, where x denotes the smallest integer not less than x. Since for any two distinct vertices u and v in UGB (n, m), the distance from u to v in the corresponding GB (n, m) provides an upper bound for the distance between u and v, we have d(UGB (n, m)) ≤ d(GB (n, m)). Therefore, the following bound is immediate. Lemma 2.1. The diameter of the generalized undirected de Bruijn graph UGB (n, m) is at most logn m. On the other hand, in UGB (n, m), the degree of every vertex is at most 2n. Therefore, from a vertex u of degree 2n − 1, one can reach at most (2n − 1) + (2n − 1)2 + · · · + (2n − 1)d vertices via a path of length d. With this observation, we get the following lower bound for the diameter. Lemma 2.2. log2n−1 m ≤ d(UGB (n, m)) for n + 1 ≤ m. Corollary 2.3. n2 ≤ m ≤ n3 .. The diameter of UGB (n, m) is 2 or 3 for. Proof. By Lemma 2.1 and log2n−1 n2 = 2 for n > 2, we ■ have the result. Now, we are ready to show our main results. Theorem 2.4. For positive integers n ≥ 2 and n3 , the diameter of UGB (n, m) is 3.. 2n2. ≤m≤. Proof. Let [0, m − 1] be the vertex set of G = UGB (n, m). We claim that either dG (0, m − n) = 3 or dG (0, m − n − 1) = 3. For convenience, let j1 = m − n and j2 = m − n − 1. By inspection, we have j1 ∈ N(0) and j2 ∈ N(0). Therefore, it suffices to prove that either N(0) ∩ N(j1 ) = ∅ or N(0) ∩ N(j2 ) = ∅, which implies that d(G) ≥ 3. Then, by Corollary 2.3, the result follows. By definition, N(0) = R(0) ∪ L(0) and N(j) = R(j) ∪ L(j) where j = j1 or j2 as the case may be. Therefore, it is equivalent to show that [R(0) ∪ L(0)] ∩ [R(j) ∪ L(j)] = ∅. We split the proof into four cases with the first three cases dealing with j = j1 or j2 . Case 1. R(0) ∩ L(j) = ∅. Since i∈R(0) R(i) = 2 − 1], neither j nor j are in R(i) = [n, n 1 2 i∈[1,n−1] i∈R(0) R(i). This implies that R(0) ∩ L(j) = ∅. Case 2. R(0) ∩ R(j) = ∅. By the definition of R(j), R(j) = {jn + α (mod m) : α ∈ [0, n − 1]}. Hence, it is clear that R(0) ∩ R(j) = ∅.. Case 3. L(0) ∩ L(j) = ∅. Assume that L(0) ∩ L(j) = ∅. Then there exists a k such that 0 ∈ R(k) and j ∈ R(k). This implies that there exist α and β where 0 ≤ α, β ≤ n − 1 satisfying kn + α ≡ 0 (mod m), (2.1) kn + β ≡ j (mod m). Therefore, β−α ≡ j (mod m) and −(n−1) ≤ β−α ≤ n−1. Since β − α = j if β − α ≥ 0 and (−β + α) + m − n < m or (−β + α) + m − n − 1 < m, we conclude that no solution (α, β) exists for (2.1). Hence the case is proved. Case 4. L(0) ∩ R(j) = ∅, j = j1 or j2 . First, we define δ(j1 ) = 0 and δ(j2 ) = 1. We claim that either 0 ∈ i∈R(j1 ) R(i) or 0 ∈ i∈R(j2 ) R(i). Assume that the above assertion is not true. Then, there exist 0 ≤ α, β, γ , ≤ n − 1 such that ((m − n − δ(j1 ))n + α)n + β ≡ 0 (mod m), ((m − n − δ(j2 ))n + γ )n + ≡ 0 (mod m). Thus,. . −n3 + αn + β ≡ 0 −n3 − n2 + γ n + ≡ 0. (mod m), (mod m).. This implies that n2 + (α − γ )n + (β − ) ≡ 0 (mod m). Since both α − γ and β − are integers between −(n − 1) and (n − 1), we have 2n2 > n2 + (α − γ )n + (β − ) > 0. Therefore, we are not able to find (α, β, γ , ) to satisfy n2 + (α − γ )n + (β − ) ≡ 0 (mod m). Hence, we conclude that either 0 ∈ i∈R(j1 ) R(i) or 0 ∈ i∈R(j2 ) R(i) and thus either L(0) ∩ R(j1 ) = ∅ or L(0) ∩ R(j2 ) = ∅. Now, combining the above four cases and j ∈ N(0), we have either dG (0, j1 ) = 3 or dG (0, j2 ) = 3. ■ 3. UGB (n, m), n 2 < m < 2n 2 Similar to Theorem 2.4, if we can find two vertices i, j ∈ [0, m − 1] such that dG (i, j) ≥ 3, then we can show that d(G) ≥ 3. First, we find the diameter of UGB (n, n2 + 1). Proposition 3.1.. d(UGB (n, n2 + 1)) = 3 for n ≥ 4.. Proof. Let m = n2 + 1 and n ≥ 4. Consider i = n − 2 and j = n2 − n + 2 in G = UGB (n, m). We claim dG (i, j) ≥ 3. Since (n2 − n + 2)n + α ≡ n + 1 + α (mod m) > n − 2 = i and (n − 2)n + α ≤ n2 − n − 1 < n2 − n + 2 = j, i ∈ R(j) and j ∈ R(i) follow. Hence, it suffices to show that [R(i) ∪ L(i)] ∩ [R(j) ∪ L(j)] = ∅ which can be broken down into four cases. • R(i) ∩ R(j) = ∅ Since (n − 2)n + α ≡ (n2 − n + 2)n + β (mod m), α − β ≡ 3n + 2 (mod m). Clearly, there are no solutions for α and β when n ≥ 4. • R(i) ∩ L(j) = ∅ Since (ni + α)n + β ≡ j (mod m), we have αn + β ≡ i + j = n2 (mod m). By the fact |αn + β| ≤ n2 − 1, there are no solutions for α and β.. NETWORKS—2008—DOI 10.1002/net. 181.

(3) • L(i) ∩ R(j) = ∅ Since (nj + α)n + β ≡ i (mod m), we have αn + β ≡ n2 (mod m) and we are not able to find solutions for α and β. • L(i) ∩ L(j) = ∅ Suppose not. Then there must exist k ∈ [0, n2 ] satisfying kn + α ≡ i (mod m) and kn + β ≡ j (mod m). Therefore, |α − β| = |i − j| = |n2 − 2n + 4| > n − 1. Again, this is not possible.. (n, n2. We note here that d(UGB + 1)) = 2 for n = 2, 3. To show the diameter of UGB (n, m) is equal to 2 for some n2 < m < 2n2 , we have to make sure that for each pair of vertices i and j, N(i) ∩ N(j) = ∅ or i ∈ N(j). Surprisingly, if m = n2 + 2, then the diameter of UGB (n, m) is equal to 2. Proposition 3.2.. d(UGB (n, n2 + 2)) = 2 for n ≥ 3.. Proof. Let m = n2 + 2. For any two distinct vertices x and y in UGB (n, m), we claim that dG (x, y) ≤ 2. It suffices to show that N(x) ∩ N(y) = ∅. Since N(x) = R(x) ∪ L(x) and N(y) = R(y) ∪ L(y), we have to prove that one of the following four conditions holds: (1) R(x) ∩ L(y) = ∅, (2) R(y) ∩ L(x) = ∅, (3) R(x) ∩ R(y) = ∅ or (4) L(x) ∩ L(y). = ∅. Observe that R(x) ∩ L(y) = ∅ if and only if (nx + α)n + β ≡ y (mod m) for some 0 ≤ α, β ≤ n − 1. Therefore, y + 2x ≡ αn + β ∈ [0, n2 − 1] (mod m). In fact, {αn + β : 0 ≤ α, β ≤ n − 1} = [0, n2 − 1]. This implies that if y + 2x ∈ [0, n2 − 1] (mod m), then d(x, y) ≤ 2. On the other hand, by considering R(y) ∩ L(x) = ∅, we have that if x + 2y ∈ [0, n2 − 1] (mod m), then d(x, y) ≤ 2. So, assume x + 2y and 2x + y are equal to either n2 or 2 n + 1 (mod m). Since 0 ≤ x = y ≤ n2 + 1, there are only six possible cases to consider. But, if 2x +y = n2 and 2y+x = 2n2 +2, then 3n2 +2 ≡ 0 (mod 3) which is not possible. By the same reason, 2x + y = n2 + 1 and 2y + x = 2n2 + 3 are not possible. Furthermore, if 2x +y = n2 and 2y +x = 2n2 +3, then y −x = n2 +3, which is not possible, either. Thus, we have exactly three cases to check. • 2x + y = n2 and 2y + x = n2 + 1 In this case, since 2n2 +1 ≡ 0 (mod 3), we may let n = 3p+1. Then x = 3p2 + 2p and y = 3p2 + 2p + 1. Hence, we have a path < 3p2 + 2p, p, 3p2 + 2p + 1 > from x to y, which concludes the proof. • 2x + y = n2 + 1 and 2y + x = 2n2 + 2 We have x = 0 and y = n2 + 1. Therefore, the path < 0, n, n2 + 1 > connects x and y for n ≥ 3, giving the result. • 2x + y = 2n2 + 2 and 2y + x = 2n2 + 3 Since 4n2 + 5 ≡ 0 (mod 3), it suffices to consider the cases n ≡ 1, 2 (mod 3). First, if n = 3p+1, then let x = 6p2 +4p+2. 182. NETWORKS—2008—DOI 10.1002/net. and y = 6p2 + 4p + 1. It is easy to see that < 6p2 + 4p + 1, 2p, 6p2 + 4p + 2 > is a path from x to y. If n = 3p + 2, the proof follows by letting x = 6p2 + 8p + 4 and y = 6p2 + 8p + 3. ■. Acknowledgments The authors are grateful to the referees and editors for their very valuable comments and suggestions which yielded an improved version of this article.. REFERENCES [1]. [2]. [3]. [4] [5]. [6]. [7]. [8]. [9] [10]. J.D.L. Caro, L.R. Nochefranca, and P.W. Sy, On the diameter of the generalized de Bruijn graphs UGB (n, n2 +1), 2000 International Symposium on Parallel Architectures, Algorithms, and Networks (ISPAN ’00), Washington, DC, USA, December 07–07, 2000, pp. 57–63. J.D.L. Caro, L.R. Nochefranca, P.W. Sy, and F.P. Muga, II, The wide-diameter of the generalized de Bruijn graphs UGB (n, n(n + 1)), 1996 International Symposium on Parallel Architectures, Algorithms, and Networks (ISPAN ’96), Washington, DC, USA, June 12–14, 1996, pp. 334–336. J.D.L. Caro and T.W. Zeratsion, On the diameter of a class of the generalized de Bruijn graphs, 2002 International Symposium on Parallel Architectures, Algorithms, and Networks (ISPAN ’02), Washington, DC, USA, May 22–24, 2002, pp. 197–202. D.Z. Du and F.K. Hwang, Generalized de Bruijn digraphs, Networks 18 (1988), 27–38. H.E. Escuadro and F.P. Muga, II, Wide-diameter of generalized undirected de Bruijn graph UGB (n, n2 ), 1997 International Symposium on Parallel Architectures, Algorithms, and Networks (ISPAN ’97), Washington, DC, USA, December 18–20, 1997, pp. 417–420. M. Imase and M. Itoh, Design to minimize diameter on building-block network, IEEE Trans Comput C 30 (1981), 439–442. L.R. Nochefranca and P.W. Sy, The diameter of the generalized de Bruijn graph UGB (n, n(n2 + 1)), 1997 International Symposium on Parallel Architectures, Algorithms, and Networks (ISPAN ’97), Washington, DC, USA, December 18–20, 1997, pp. 421–423. S.M. Reddy, D.K. Pradhan, and J.G. Kuhl, Directed graphs with minimal diameter and maximal connectivity, School of Engineering, Oakland University Technical Report, Oakland, USA, July 1980. D.B. West, Introduction to graph theory, Prentice-Hall, Upper Saddle River, NJ, 2001. J.M. Xu, Wide diameters of cartesian product graphs and digraphs, J Combin Optim 8 (2004), 171–181..

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