Spnning tree congestion of some product grphs Hiu-Fi Lw Mthemticl Institute Oxford University 4-9 St Giles Oxford, OX1 3LB, United Kingdom e-mil: lwh@mths.ox.c.uk nd Mikhil I. Ostrovskii Deprtment of Mthemtics nd Computer Science St. John s University 8000 Utopi Prkwy Queens, NY 11439, USA e-mil: ostrovsm@stjohns.edu September 9, 010 Abstrct. We estimte spnning tree congestion for crtesin products of pths nd complete grphs. Keywords: Crtesin products, minimum congestion spnning tree, spnning tree congestion 010 Mthemtics Subject Clssifiction. Primry: 05C05 1 Introduction nd bsic definitions Let G be grph nd let T be spnning tree in G (sying this we men tht T is subgrph of G). We follow the terminology nd nottion of []. For A, B V (G) by e G (A, B) we denote the number of edges in G with one end in A nd the other end in B. For ech edge e of T let A e nd B e be the vertex sets of the components of T \e. We define the edge congestion of G in T by ec(g : T ) = mx e E(T ) e G(A e, B e ). The number e G (A e, B e ) is clled the congestion in e. The nme comes from the following nlogy. Imgine tht edges of G re rods, nd edges of T re those rods which re clened from snow fter snowstorms. If we ssume tht 1
ech edge in G bers the sme mount of trffic between its vertices, nd tht fter snowstorm ech driver tkes the corresponding (unique) detour in T, then ec(g : T ) describes the trffic congestion t the most congested rod of T. It is cler tht for pplictions it is interesting to find spnning tree which minimizes the congestion. We define the spnning tree congestion of G by s(g) = min{ec(g : T ) : T is spnning tree of G}. (1) Ech spnning tree T in G stisfying ec(g : T ) = s(g) is clled minimum congestion spnning tree. The prmeters ec(g : T ) nd s(g) were introduced nd studied in [9]. Some versions of these prmeters were introduced nd studied before, see, for exmple, [5] nd [10]. One of the interesting problems bout the spnning tree congestion is to estimte/evlute it for some stndrd fmilies of grphs. In this pper we continue the study (see [3], [4], [6], [7], nd [8]) of the spnning tree congestion of crtesin products of complete grphs nd pths. An improvement of the upper bound for 3-dimensionl cubic grids By 3-dimensionl cubic grid [k] 3 we men the crtesin cube of pth of length (k 1), tht is, the grph whose vertex set is the set of ll triples in {0, 1,..., k 1} 3, nd vertices (x 1, x, x 3 ) nd (y 1, y, y 3 ) re djcent if for some i we hve x i y i = 1, nd x j = y j for ll j i. The spnning tree congestion of [k] 3 ws studied in [3] nd [7]. The min result of [3] on this mtter cn be stted s: 6 k o(1) s([k] 3 ) 7 8 k + O(k). () The estimte from below ws independently obtined in [7]. In both ppers the estimte from below is derived from inequlities of [1]. Our purpose now is to improve the upper estimte, replcing 7 by smller 8 number. In the sme wy s in [3] this result cn be derived from the following geometric result. A brick in R 3 is set of the form 3 i=1 [ i, b i ], where i < b i for ll i. A rectiliner body is finite union of bricks. For rectiliner body A [0, 1] 3 we write σ(a) for the surfce re of A in the interior of [0, 1] 3 nd m(a) for the volume (Lebesgue mesure) of A. We denote the re of -dimensionl set S by m (S). Theorem 1. There exists prtition of the solid cube [0, 1] 3 into 6 rectiliner bodies {A i } 6 i=1 such tht mx 1 i 6 m(a i ) 1 nd mx σ(a i) = 3 + 15. (3) 1 i 6
S S 1 Figure 1: Cuts of [0, 1] [0, 1] Note. The min point of this result is tht the right-hnd side of (3) is < 7/8, simpler prtition for which the mximum in (3) is 7/8 ws found in [3]. Proof of Theorem 1. We let A 1 = [0, ] [1, 1] [0, 1], A = [1, 1] [0, ] [0, 1], where < 1. We cut the fces [0, 1] [0, 1] {0} nd [0, 1] [0, 1] {1} of the cube using rectiliner cuts shown in Figure 1 nd in such wy tht m (S 1 ) = m (S ). The remining elements of the prtition: A 3 = S 1 [ 0, 1 ], A 4 = S 1 [ 1, 1 ], A 5 = S [ 0, 1 ] [ ] 1, A 6 = S, 1. It is esy to see tht σ(a 1 ) = σ(a ) = nd tht σ(a 3 ) = σ(a 4 ) = σ(a 5 ) = σ(a 6 ). We pick in such wy tht σ(a 1 ) = = σ(a 6 ). It remins to find this number. We hve σ(a 3 ) = m (S 1 ) + (1 ) = 1 + (1 ). Equting σ(a 1 ) nd σ(a 3 ) we get = 3+ 15. 3 Crtesin products of complete grphs Spnning tree congestion of crtesin products of complete grphs ws studied in [6]. Our purpose here is to get upper estimtes for the spnning tree congestion of crtesin powers of complete grphs. Let d, n N, we introduce the grph H = H(d, n) s crtesin product of d copies of the complete grph K n. This mens tht vertices of H re d-tuples of vertices of K n ; two d-tuples 3
(v 1,..., v d ) nd (u 1,..., u d ) re djcent in H if nd only if u i = v i for ll but one i {1,..., d}. When n is cler from context we lso write H d for H(d, n). It ws shown in [6, Lemm 4.4] tht s(h) (n d 1) log n d/d for n, d 3. We use the pproch of [8] to show tht for fixed n nd lrge d ( ) logn d s(h d ) = O n d, (4) ( nd thus s(h d ) = Θ logn d n ). d d To estblish (4) we construct tree T such tht ec(h d : T ) hs suitble estimte from bove. We identify the vertex set of K n with {0, 1,..., n 1} nd denote by 0 d the d-tuple (0,..., 0). We ssume d, n 3. First we construct wht we cll stndrd spnning tree T d of H d, it is tree with centroid t 0 d nd such tht ec(h d : T d ) = (n 1)n d 1. Let T 1 be the spnning tree of H 1 with edge set {0i : i = 1,,, n 1}. Clerly, the congestion of T 1 is n 1. For j 1 nd 0 i n 1, let Tj i = T j {i} H j+1 nd S j+1 = 0 j T 1. Then T j+1 = ( n 1 i=0 T j i ) S j+1 is spnning tree of H j+1, with the property tht one of the components obtined fter deletion of ny of the edges is isomorphic to H for some 0 j. Counting edges joining such component with its complement we get ec(h j+1 : T j+1 ) = mx 0 j ((j + 1 )(n 1)n ) = (n 1)n j. Thus, we hve obtined preliminry estimte s(h d ) (n 1)n d 1. Now we describe our construction of spnning tree of lower congestion in H d for d lrge compred to n. Lter we shll lso use the observtion tht the rdius of the stndrd spnning tree is d nd tht 0 d is its centrl vertex. 1. We prtition V (H d ) = v V (Hk )V (H v ) where k = log n d(n 1) nd V (H v ) = {v} V (H d k ). In other words, we prtition H d into t most d(n 1) pieces which re isomorphic to crtesin powers of K n. The min ide behind this is tht the edge-boundry of ech such piece is close to the bound given by edge-isoperimetric inequlity; this is why we get close-to-optiml bound for congestion.. The spnning tree which we construct is union of trees S(v) which re close to stndrd spnning trees of H v nd tree F, which we cll frme, connecting H v with 0 d ; 0 d is centroid of the constructed tree. The tree F is constructed s subdivided str with V (H k ) leves, ech leve is in the corresponding V (H v ). 3. The tree F in ddition to lef in V (H v ) hs to contin some vertices of v V (H k ) V (Hv ) s vertices of degree. For this reson the union of F nd of ll stndrd spnning trees in H v is not tree, we hve to modify stndrd spnning trees in H v nd to show tht this modifiction does not ffect much the spnning tree congestion. d 4
Since we ssumed d, n 3, we hve k = log n d(n 1) 1. We turn to construction of F. Let T be stndrd spnning tree of H k. Since the degree of 0 k in T is p := k(n 1), we my choose leves l 1, l,, l p such tht the pths in T from 0 k to l 1, l,, l p re edge-disjoint. We construct pth in H d joining 0 d nd H l i (i = 1,,, p) in the following wy. Let 0 k T l i = 0 k v i1 v i l i be the pth joining 0 k nd l i in T. Then 0 d (v i1, 0 d k )(v i, 0 d k ) (l i, 0 d k ) is the desired pth, we denote it P (l i ). Let r 1 = 0 k, r,, r q be the remining vertices of T, where q = n k k(n 1) (n 1)(d k). Let ϕ : {r 1, r,, r q } {(, b) {1,..., d k}, b {1,..., n 1}} be n injection such tht ϕ(r 1 ) = (1, 1). For ech 1 j q, let f j = be where ϕ(r j ) = (, b) nd {e 1, e,, e d k } forms unit vector bsis of R d k. We identify f j with the corresponding vertex in H d k. We construct pths P (r j ) joining 0 d nd H r j s follows. We let P (r 1 ) = 0 d (0 k, f 1 ). If j nd 0 k T r j = 0 k u j1 u j r j is the corresponding pth in T, we let P (r j ) = 0 d (0 k, f j )(u j1, f j )(u j, f j ) (r j, f j ). The union of ll pths P (l i ) nd P (r j ) is subdivided str which is the desired frme F. We use the term connecting vertices for leves of F. Our next step is to find spnning tree S(v) in the reminder of ech H v. By the reminder we men H v from which we remove ll vertices hving degree in F. If v = l i, there re no vertices of F of degree in H v, nd we let S(v) to be stndrd spnning tree in H v with the centroid t the connecting vertex (v, 0 d k ). For v = r j the connecting vertex is neighbor of (v, 0 d k ), becuse f j hs only one non-zero coordinte. Some other neighbors of (v, 0 d k ) lso cn hve degree on F, they cn belong to some of the pths P (r k ), k j. The vertex (v, 0 d k ) itself cn belong to one of P (l i ). Since H d k is edge-trnsitive, it is esy to check tht there is n isomorphism of H d k onto H v which mps the stndrd spnning tree of H d k onto tree R(v) for which ll neighbors of (v, 0 d k ) re leves. Deleting from R(v) those of these leves which hve degree in F we get the desired tree S(v). Similr construction works for r 1 = 0 k nd f 1 = e 1. An importnt observtion which follows from our definition of f j is tht every edge of S(v) disconnects either copy of H, or copy of H with one vertex deleted for 0 d k 1. We form the desired spnning tree in H d s T = ( v H k S(v) ) F. Now we estimte ec(h d : T ). The congestion in n edge of S(l i ) which seprtes component isomorphic to H ( {1,..., d k 1}) is (n 1)(d )n. This quntity is mximized when = d k 1, so it does not exceed (n 1)(k + 1)n d k 1. 5
The congestion in n edge of S(r j ), which seprtes component isomorphic to H cn be estimted s bove. If the seprted component is H with one vertex deleted, it is esy to see tht the congestion does not exceed (n 1)(d )(n 1) + (n 1). As the derivtive of this function with respect to is positive, we conclude tht the congestion does not exceed (n 1) ( (k + 1)n d k 1 + d ). The congestion in n edge of F incident with connecting vertex of H v cn be estimted s follows. Let b {1,..., q} be the number of other vertices in V (H v ) V (F). The congestion does not exceed (n 1)k(n d k b) + b(n 1)(d k) < (n 1)(kn d k + bd) (n 1)[kn d k + (n k k(n 1))d] The length of the pth from the connecting vertex to 0 d in F does not exceed k + 1. The congestion increses by t most d(n 1) with ech edge on the wy from the connecting vertex to 0 d (becuse d(n 1) is the degree of vertex). Therefore, the congestion in n edge in F is t most (n 1) ( kn d k + (n k k(n 1))d + (k + 1)d ). Tking the mximum of these three estimtes we get ec(h d : T ) (n 1)[kn d k + d(n k kn + k + 1)]. Thus, we hve proved the following upper bound. As we observed bove, if n is fixed nd d is lrge, this bound is close to the corresponding lower bound from [6]. Theorem. If d, n 3, then {( ) } logn d(n 1) s(h d ) (n 1) min n d + d (n 1), n d 1. n log n d(n 1) References [1] B. Bollobás nd I. Leder, Edge-isoperimetric inequlities in the grid, Combintoric, 11 (1991), no. 4, 99 314. [] J. A. Bondy, U. S. R. Murty, Grph theory, Grdute Texts in Mthemtics, 44, Springer, New York, 008. [3] A. Cstejón, M. I. Ostrovskii, Minimum congestion spnning trees of grids nd discrete toruses, Discussiones Mthemtice Grph Theory, 9 (009), 511 519. [4] S. W. Hrusk, On tree congestion of grphs, Discrete Mth., 308 (008), 1801 1809. [5] S. Khuller, B. Rghvchri, N. Young, Designing multi-commodity flow trees, Informtion Processing Letters, 50 (1994), 49 55. [6] K. Kozw, Y. Otchi, On spnning tree congestion of Hmming grphs, preprint, 009. [7] K. Kozw, Y. Otchi, K. Ymzki, On spnning tree congestion of grphs, Discrete Mth., 309 (009), 415 44. [8] H.-F. Lw, Spnning tree congestion of the hypercube, Discrete Mth., 309 (009), 6644 6648. 6
[9] M. I. Ostrovskii, Miniml congestion trees, Discrete Mth., 85 (004), 19 6. [10] S. Simonson, A vrition on the min cut liner rrngement problem, Mth. Systems Theory, 0 (1987), no. 4, 35 5. 7