On spanning tree congestion of Hamming graphs

arXiv:1110.1304v3 [cs.DM] 17 Oct 2011 On spanning tree congestion of Hamming graphs Kyohei Kozawa∗ Yota Otachi† November 4, 2018 Abstract We present a tight lower bound for the spanning tree congestion of Hamming graphs. 1 Preliminaries The spanning tree congestion of graphs has been studied intensively [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. In this note, we study the spanning tree congestion of Hamming graphs. We present a lower bound for the spanning tree congestion of Hamming graphs. That is, in our terminology, we show that stc(Kd n) ≥1 d(nd −1) logn d. It is known that stc(Kd n) = O  1 dnd logn d  [10]. Thus our lower bound is asymptotically tight. For a graph G, we denote its vertex set and edge set by V(G) and E(G), respectively. For S ⊆V(G), let G[S ] denote the subgraph induced by S . For an edge e ∈E(G), we denote by G −e the graph obtained from G by deleting e. Let NG(v) denote the neighborhood of v ∈V(G) in G; that is, NG(v) = {u | {u, v} ∈E(G)}. We denote the degree of a vertex v ∈V(G) by degG(v), and the maximum degree of G by ∆(G); that is, degG(v) = |NG(v)| and ∆(G) = maxv∈V(G) degG(v). A graph G is r-regular if degG(v) = r for every v ∈V(G). For S ⊆V(G), we denote the edge set of G[S ] by ιG(S ), and the boundary edge set by θG(S ); that is, ιG(S ) = {{u, v} ∈E(G) | u, v ∈S } and θG(S ) = {{u, v} ∈E(G) | exactly one of u, v is in S }. We define the function ι and θ also for a positive integer s ≤|V(G)| as ιG(s) = maxS ⊆V(G), |S |=s |ιG(S )| and θG(s) = minS ⊆V(G), |S |=s |θG(S )|. Let T be a spanning tree of a connected graph G. The conges- tion of e ∈E(T) as cngG(e) = |θG(Le)|, where Le is the vertex set of one of the two components of T −e. The congestion of T in G, denoted by cngG(T), is the maximum congestion over all edges in T. We define the spanning tree congestion of G, denoted by stc(G), as the minimum congestion over all spanning trees of G. The d-dimensional Hamming graph Kd n is the graph with vertex set {0, . . . , n −1}d in which two vertices are adjacent if and only if their corresponding d-dimensional vectors differ in exactly one place. It is evident that Kd n is d(n −1)-regular. The exact value of stc(K2 n) is known [6]. Also, stc(Kd 2) is determined asymptotically [8]. ∗Electric Power Development Co., Ltd., 6-15-1, Ginza, Chuo-ku, Tokyo, 104-8165 Japan. †Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan. E-mail address: otachi@dais.is.tohoku.ac.jp 1 2 The lower bound Here, we present the lower bound. We need the following three lemmas. Lemma 2.1 ([1]). If G is r-regular and S ⊆V(G), then |θG(S )| = r|S | −2|ιG(S )|. Lemma 2.2 ([17]). Let G be a subgraph of Kd n. If G has s vertices and t edges, then 2t ≤(n − 1)s logn s. Lemma 2.3 ([4, 7]). For any connected graph G, stc(G) ≥min⌊|V(G)|/2⌋ s=⌈(|V(G)|−1)/∆(G)⌉θ(s). Theorem 2.4. stc(Kd n) ≥(nd −1) logn d/d for n, d ≥3. Proof. Since Kd n is d(n −1)-regular, Lemmas 2.1 and 2.2 imply that θKdn(s) ≥(n −1)s(d −logn s). Let f (s) = (n −1)s(d −logn s) and f ′(s) be the derived function of f (s). Then f ′(s) = (n −1)(d − 1/ ln n −logn s), and thus, f (s) is increasing for (nd −1)/(d(n −1)) ≤s ≤nd−1/ ln n and decreasing for nd−1/ ln n ≤s ≤nd/2. Therefore, ⌊nd/2⌋ min s=⌈(nd−1)/(d(n−1))⌉ f (s) = min ( f & nd −1 d(n −1) '! , f $nd 2 %!) ≥min ( f nd −1 d(n −1) ! , f nd 2 !) = min (nd −1 d

d −logn nd −1 d(n −1) ! , (n −1)nd 2

d −logn nd 2 !) ≥min (nd −1 d logn d, (n −1)nd 2 logn 2 ) . Thus, by Lemma 2.3, it holds that stc(Kd n) ≥min (nd −1 d logn d, (n −1)nd 2 logn 2 ) . By a simple calculation, we can see that nd−1 d logn d ≤(n−1)nd 2 logn 2 for d = 2, 3. Since nd −1 ≤ (n −1)nd and (logn d)/d ≤(logn 2)/2 for d ≥4, the theorem holds. □ References [1] S. L. Bezrukov, Edge isoperimetric problems on graphs, in: L. Lovász, A. Gyárfás, G. O. H. Katona, A. Recski, L. Székely (eds.), Graph Theory and Combinatorial Biology, vol. 7 of Bolyai Soc. Math. Stud., János Bolyai Math. Soc., Budapest, 1999, pp. 157–197. [2] H. L. Bodlaender, F. V. Fomin, P. A. Golovach, Y. Otachi, E. J. van Leeuwen, Parameterized complexity of the spanning tree congestion problem, Algorithmica. 2 [3] H. L. Bodlaender, K. Kozawa, T. Matsushima, Y. Otachi, Spanning tree congestion of k- outerplanar graphs, Discrete Math. 311 (2011) 1040–1045. [4] A. Castejón, M. I. Ostrovskii, Minimum congestion spanning trees of grids and discrete toruses, Discuss. Math. Graph Theory 29 (2009) 511–519. [5] S. W. Hruska, On tree congestion of graphs, Discrete Math. 308 (2008) 1801–1809. [6] K. Kozawa, Y. Otachi, Spanning tree congestion of rook’s graphs, Discuss. Math. Graph Theory 31 (2011) 753–761. [7] K. Kozawa, Y. Otachi, K. Yamazaki, On spanning tree congestion of graphs, Discrete Math. 309 (2009) 4215–4224. [8] H.-F. Law, Spanning tree congestion of the hypercube, Discrete Math. 309 (2009) 6644–6648. [9] H.-F. Law, M. I. Ostrovskii, Spanning tree congestion: duality and isoperimetry; with an application to multipartite graphs, Gr

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