ON THE EDGE-BANDWIDTH OF GRAPH PRODUCTS

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1 ON THE EDGE-BANDWIDTH OF GRAPH PRODUCTS JÓZSEF BALOGH, DHRUV MUBAYI, AND ANDRÁS PLUHÁR Abstract The edge-badwidth of a graph G is the badwidth of the lie graph of G We show asymptotically tight bouds o the edge-badwidth of two dimesioal grids ad tori, the product of two cliques ad the -dimesioal hypercube 1 Itroductio Let G = (V (G), E(G)) be a simple graph with vertices A labellig η is a bijectio of V (G) to {1,, } The badwidth of η is The badwidth B(G) of G is B(η, G) = max{ η(u) η(v) : uv E(G)} B(G) := mi{b(η, G)} η The otio first came up i the semial paper of Harper [6] i which the badwidth of the -dimesioal hypercube was give It turs out that the determiatio or computatio of the badwidth of graphs is hard (i fact, it is NP-hard [14]); for a good survey, see [4] or [1] The edge-badwidth was itroduced by Hwag ad Lagarias i [9] Here the edges are labelled istead of the vertices, ad the badwidth of a edge-labellig η of a graph G is B (η, G) := max{ η(uv) η(vw) : uv, vw E(G)} The edge-badwidth of a graph G is B (G) := mi{b (η, G)} η Of course B (G) = B(L(G)), where L(G) is the lie graph of G, see [10] The Cartesia product of graphs G ad H is deoted by G H, with V (G H) = {(u, v) u V (G), v V (H)} ad E(G H) = { (u 1, v 1 ), (u, v ) u 1 = u, (v 1, v ) E(H) or (u 1, u ) E(G), v 1 = v } The -th fold product G G G is deoted G I this paper we shall give estimates for edge-badwidth of four types of graph 1991 Mathematics Subject Classificatio 05C78 Key words ad phrases Badwidth, edge-badwidth, lie graph, hypercube, grid, isoperimetric problems The first authors research is partially supported by NSF grat DMS The secod authors research is partially supported by NSF grat DMS The third author was partially supported by OTKA grat T

2 BALOGH, MUBAYI, AND PLUHÁR products, P P, where P deotes the path with vertices, C C, where C is the cycle o vertices, K K, where K is the clique o vertices, ad P = K, the -dimesioal hypercube The badwidths of P P, C C ad K K are well studied, the first oe is, the secod is 1, while the third oe is ( + 1)/, see [5, 13, 1] Nevertheless, oly the trivial lower bouds o B (P P ) ad ( 1)/3 B (K K ) are kow o the edge-badwidth of those (Oe ca readily get those by Propositio 5, i the ext sectio) Note that it is easy to see that B (P ) = 1, B (C ) =, ad i [10] it was proved that B (K ) = /4 + / ad B (K, ) = ( ) +1 1 where K, deotes the complete bipartite graph Our results are the followig Theorem 1 Let The (1) 1 B (P P ) 1 ad () 4 1 B (C C ) 4 1 We also obtai asymptotically tight bouds o the edge-badwidth of the product of two equal cliques Theorem (3) B (K K ) The third family of graphs have bee studied extesively earlier Recall that P is the -dimesioal hypercube, that is the vertices of P are the 0-1 sequeces of legth, ad there is a edge betwee the vertices x ad y iff their Hammig distace is oe Bezrukov, Grüwald ad Weber [] showed that ( ) 1 + B (P ) 1 A improved lower boud o B (P ) was proved by Calamoeri, Massii ad Vrťo [3], amely that ( ) 4 B (P ) Our fial result establishes the right asymptotical growth of B (P ): Theorem 3 (4) B (P ) = ( ) ( ) + o()

3 EDGE BANDWIDTH 3 Geeral Bouds The stadard techiques for obtaiig lower bouds o badwidth apply isoperimetric iequalities I the literature may vertex ad edge isoperimetric problems were cosidered, i particular o the square grid ad the hypercube Give a graph G, for a S V (G) let (S) = {v V (G) \ S (u, v) E(G), u S} A typical (vertex) isoperimetric questio is that for a give graph G of order, ad for a fixed iteger k what is L k (G) := mi (S)? S =k, S V (G) As Propositio 4 states, the value of max k L k (G) is a lower boud for B(G) It is ot hard to see that this boud is sharp if the extremal structures for differet k s, achievig the isoperimetric boud, ca be positioed i G to be built a ested sequece of sets, more precisely a sequece {S k } k=1 V (G) ca be build that for all i < j, S i S j ad either S i (S i ) S j or S j S i (S i ) See [7] for more details I our cases, Propositio 4 does ot eve give asymptotically sharp bouds, but surprisigly the iterated versio of it, Propositio 6 does I particular, we remark that the vertex isoperimetric umber of L(P P ) + 1, but the badwidth is aroud Propositio 4 [6] Let G be a graph ad k be a iteger, 0 k V (G) The B(G) mi max{ (S), (V S) } S, S =k The legth of a P is 1 The distace of two vertices i a graph G is the legth of the shortest path betwee them, ad the diameter, diam(g), of a graph G is the maximum distace betwee its vertices Propositio 5 [4] Let G be a graph The B(G) V (G) 1 diam(g) Sice we eed a extesio of these results, we give the outlie of their proofs Fix a labellig of G, ad let S be the set of vertices labelled by the umbers {1,, k} Now the largest umber appearig o the vertices i (S) is at least k + (S), which gives a (absolute) differece at least (S) with the label of some vertex of S; that is B(G) (S) Usig the same estimate for V S ad takig the optimal labellig, Propositio 4 follows For Propositio 5 cosider a shortest path coectig the vertices labelled by 1 ad V (G) The average (absolute) differece betwee the labels of eighborig vertices is at least ( V (G) 1)/diam(G), hece the largest differece is at least this much

4 4 BALOGH, MUBAYI, AND PLUHÁR Similarly to Propositio 4, oe may cosider ot oly the set (S), but l (S) := ( l 1 (S)) l i=1 i 1 (S), where 0 (S) := S Let σ l (S) := l i=1 i (S), shortly the l-shadow of S Note that every vertex of σ l (S) is coected to S by a path of legth at most l As before, fix a labellig of G, ad let S be the set of vertices labelled by the umbers {1,, k} The biggest label i σ l (S) is at least k + σ l (S), say this appears o vertex y σ l (S) Cosider ow a shortest path coectig a arbitrary vertex x of S ad vertex y The average (absolute) differece betwee the labels of eighborig vertices is at least σ l (S) /l, hece the largest differece is at least this much This yields the followig result Propositio 6 Let G be a graph ad k a iteger, 0 k V (G) The we have (5) B(G) mi max S, S =k 1 l σ l (S) l 3 The proof of Theorem 1 31 The case of grids We eed to prove that 1 B (P P ) 1 Here we have two cadidates for optimal labelligs that are differet from each other, which perhaps makes fidig B (P P ) harder, see Figure 1 The vertices of P P are labelled with (i, j) where 1 i, j Labellig 1 Let ad η( (i, j), (i, j + 1) ) := (i 1)( 1) + j η( (i, j), (i + 1, j) ) := i( 1) + j Labellig We cosider the vertices of the grid as the elemets of a matrix For edges bellow the diagoal (1, ) (, 1) let ad η( (i, j), (i, j + 1) ) := (i + j )(i + j 1) + i 1 η( (i, j), (i + 1, j) ) := (i + j )(i + j 1) + i Otherwise we exted the labels i atisymmetric way: ad η( (i, j), (i, j + 1) ) := + 1 η( + 1 i, + 1 j, + 1 i, j ) η( (i, j), (i + 1, j) ) := + 1 η( + 1 i, + 1 j, i, + 1 j ) It is ot hard to check that the badwidth of both labelligs is 1

5 EDGE BANDWIDTH Labellig 1 for = 4 Labellig for = 4 Figure 1 For the proof of the lower boud, we eed the followig Lemma Lemma 7 (i) Let G be the tree cosistig of a path of legth 1 together with 1 additioal edges icidet with all but the first vertex of the path (So V (G) = 1 ad E(G) = ) Let D be a oempty set of edges i G Suppose that E(G) D r 1 for some r > 0 The σ r (D) r 1 (ii) Let H be the graph cosistig of a cycle of legth, together with additioal edges icidet with all vertices of the cycle (So V (H) = = E(H) ) Suppose that E(H) D 4r for some r > 0 The σ r (D) 4r Proof (i) Defie the distace d(e, f) betwee two edges e ad f to be oe less tha the legth of the shortest path startig with e ad edig with f For disjoit sets of edges P, Q, defie the distace d(p, Q) betwee P ad Q to be the miimum, over all edges e P ad f Q of d(e, f) Amog all edge sets T E(G) D of size r 1, cosider the oe that miimizes e T d({e}, D) Call this set T 0 Note that T 0 exists sice by our hypothesis E(G) D r 1 We will show that T 0 σ r (D) This suffices to complete the proof, sice T 0 = r 1 Suppose, o the cotrary, that there exists a edge e T 0 such that d({e}, D) r + 1 Let P be a shortest path startig with e ad edig i D, say at edge f D The the legth of P is at least r + Let X be the set of r edges

6 6 BALOGH, MUBAYI, AND PLUHÁR of P {f} closest to f (ote that e X) By the defiitio of G, there is a set Y of r 1 edges outside of P, icidet to X, ad icidet to either e or f (thus e Y ) Because of the edges of P, oe ca see that X Y σ r (D) By the defiitio of T 0, X Y T 0, sice otherwise we could replace e by a edge from X Y, cotradictig the miimality of T 0 Now X Y = r 1 ad e T 0 (X Y ) leads to T 0 > r 1, a cotradictio (ii) We use a similar strategy to prove this part as i part (i) Fix a edge set D such that = D E(H) ad E(H) D 4r Defie T 0 as i (i) Agai, to get a cotradictio we have to cosider a edge e such that d({e}, D) r + 1 Now we may fid two edge disjoit shortest paths from e to D Let P 1 deote a shortest path from e to D ot cotaiig e (This path has legth at least 1 sice r > 0) Deletig the edges of P 1, the edge e is still i a compoet with some edges of D, hece there exists a shortest path P coectig them, which is edge disjoit from P 1 Now repeatig the argumet for both paths that we did i (i) for P, we ca obtai the statemet (ii) We cosider the vertices of P P as the elemets of a matrix, thus the vertex (i, j) lies i row i ad colum j Call a edge horizotal if it is of the form (i, j), (i, j + 1), ad vertical if it is of the form (i, j), (i + 1, j) The left (right) vertex of (i, j), (i, j + 1) is (i, j) ((i, j + 1)), ad the top (bottom) vertex of (i, j), (i + 1, j) is (i, j) ((i + 1, j)) Defie the row r i (colum c j ) to be the set of 1 horizotal (vertical) edges (i, 1), (i, ), (i, ), (i, 3),, (i, 1), (i, ) ( (1, j), (, j), (, j), (3, j),, ( 1, j), (, j) ) Defie a lie to be a row or a colum Let t be the smallest iteger such that edges labelled 1,, t cotai a lie l Assume without loss of geerality that l is a row that is ot r 1 Let S be the set of edges labelled by 1,, t Defie R to be the set of rows r for which 1) there is a vertical edge from S whose bottom vertex is o r, or ) there is a horizotal edge from S i r Claim 1 (S) + R 1 Proof By the defiitio of t, (S) cotais oe vertical edge from each colum By the defiitio of R, (S) cotais oe horizotal edge from each row i R except l This gives vertical edges ad R 1 horizotal edges i (S) Claim σ R (S) ( 1)( R ) Proof We begi with by associatig to each colum c j (for j > 1), a set E j of ( R ) 1 edges, such that E j E j = for j j For each vertex (i, j) for which r i R ad i > 1, we cosider the two edges e = (i, j 1), (i, j) ad f = (i 1, j), (i, j) I other words, these edges are the horizotal edge with right edpoit (i, j) ad the vertical edge with bottom edpoit (i, j) By defiitio of R, either e or f are i S

7 EDGE BANDWIDTH 7 This way we get at least ( R ) 1 edges outside S (we get oe more edge if r 1 R) Now cosider the graph G j cosistig of all 1 vertical edges of the colum c j for a fixed j = 1,,, ad the 1 horizotal edges (i, j 1), (i, j), where i > 1 Let D = S E(G) The G j satisfies the hypothesis of Lemma 7 with r = R, (ote that D because of the row l) ad we coclude that ( R ) 1 of the edges of G j lie i σ R (D) For the colum c 1 a similar argumet produces R 1 vertical edges i σ R (S) Altogether we have produced ( 1)(( R ) 1)+ R 1 = ( 1)( R ) differet edges i σ R (S) Now if R, the Claim 1 ad Propositio 4 apply, while if R, the Claim ad Propositio 6 apply Puttig these together, we get B (P P ) mi max R { + R 1, ( 1)( R ) R } 1 3 The case of tori We leave the easy costructio for the upper boud to the reader The proof of the lower boud is very similar to the proof of (1) Cosider a edge labellig of C C Let t be the smallest iteger such that the set of edges labelled by 1,,, t cotais all but oe edges of a lie l Without loss of geerality, l is a row Let agai R be the set of rows r for which there is a vertical edge from S whose bottom vertex is o r, or there is a horizotal edge from S i r Claim 1 (S) + R 1 Proof By defiitio of t, (S) cotais at least two vertical edges from each of the colums ad from each row i R except l, ad 1 from l Claim σ ( R )/ (S) ( R ) Proof We begi with by associatig to each colum c j a graph H j (isomorphic to the oe of Lemma 7 (ii)), ad let D := S E(H j ) The E(H j ) D ( R ), by the defiitio of R This meas that Lemma 7 (ii) ca be applied, with r = ( R )/, provig the statemet (Note that we ca always assume that the itersectio of the row l ad H j is i S, providig that D is ot empty) Now if R, the Claim 1 ad Propositio 4 apply, ad if R <, the Claim ad Propositio 6 apply 4 Product of two cliques First we demostrate that B (K K ) 3 3 /8 + 9 /8 To simplify the costructio, we give a mappig η : E(K K ) [1,, 3 ] istead of mappig the edges oto [1,, ( ) ] A further simplificatio is that we shall ot bother with the error terms of quadratic sizes, divisibility or the exact edpoits of the subitervals of [1,, 3 ] Before gettig ito the quite paiful details, let us outlie the ideas behid the costructio We cosider the vertices of K K as cells of a matrix, ad the edges are amog the cells of a row or colum Obviously, if we put the smallest

8 8 BALOGH, MUBAYI, AND PLUHÁR umbers to the upper left part, the the biggest umbers have to be placed to the lower right part So the first idea oe may thik is to divide the matrix ito four equal sub-matrices (upper, lower, left ad right), ad use up the umbers for the edges i the followig order Fill first the upper left sub-matrix, the the edges betwee the upper ad lower left sub-matrices, the the lower sub-matrix ad so o However, oe rus ito great difficulties whe tryig to decide about the labels of edges goig betwee the left ad right side To overcome these difficulties we use a trickier divisio of the matrix, ad usig the umbers for labellig more ecoomically This meas to save some of the smaller umbers, ad use those up oly later, where the aive costructio would result i too big differeces We also have to maitai a symmetry i order to keep the umber of appearig cases reasoably small We start with explaiig this symmetry first, the the labellig of edges iside a sub-matrix, fially the divisio ad the labels amog those matrices The costructio ivolves some optimizatio, that is why we had to defie some strage lookig umbers Cuttig up this matrix ito rectagles, the fuctio η shall be defied o the edges iside rectagles ad betwee two rectagles A rectagle will be specified by its upper left ad lower right corer The fuctio η will be atisymmetric with respect to the ceter of the matrix, that is ad η( ( i + 1, j + 1), ( k + 1, j + 1) ) = 3 η( (i, k), (i, j) ) η( ( i + 1, j + 1), ( i + 1, k + 1) ) = 3 η( (i, j), (i, k) ) This way it suffices to defie η for oly half of the edges The first method is to assig labels from a give set of umbers I to all edges of a l by k rectagle T called the simple block The elemets of I are used i order, startig from the smallest to fill T row by row That is for every i, assumig that the edges (iside) of a i by k sub-rectagle T i are labelled, the the edges coectig the vertices of the (i + 1)st row with the vertices of T i are labelled, fially the edges iside the (i + 1)st lie get their label I the first case we proceed row by row, like readig a text, ad order the edges coected to (i + 1, j) by the first coordiate of their other edpoit The secod oe is doe i a lexical way accordig to the secod coordiates of the edpoits, i e the order of the labellig is (i + 1, 1), (i + 1, ), (i + 1, 1), (i + 1, 3), (i + 1, ), (i + 1, 3), ad so o Let a := ( 1)/4 We shall refer to the followig sub-rectagles: T (1) with corers (1, 1) ad ( a, /), T () with corers ( a + 1, 1) ad (, /), T (3) with corers (1, 1) ad (/4, /), T (4) with corers (/4 + 1, 1) ad (/, /), T (5) with corers (/ + 1, 1) ad (3/4, /)

9 EDGE BANDWIDTH 9 T (6) with corers (3/4 + 1, 1) ad (, /) Let furthermore T (i) be the cetrally symmetric image of T (i) for i = 1,, 6 T(1) T (1) T () T(3) T (6) T(4) T (5) T(5) T (4) T(6) T (3) T() The rectagles Figure First we use the iterval [1,, ( a)(3/ a)/4] to make a simple block out of T (1) Next we use the iterval [( a)(3/ a)/4 + 1, 3 3 /64] to label the edges betwee T (1) ad T () The order is the same that we used i buildig the simple block, but there are o edge labels iside the rows ow The iterval [3 3 /64, 7 3 /64] is used to label the edges betwee T (3) ad T (6) It is doe similarly as before (goig through the rows of T (6), ad order the edges by the secod coordiate of their other edpoit) The most subtle part is the labellig of the edges betwee T (4) ad T (5) Now the labels are from the iterval [7 3 /64 + 1, 3 3 /64] There are 4 3 /64 edges to be

10 10 BALOGH, MUBAYI, AND PLUHÁR labelled, that is 3 /64 umbers will be saved for later use For a edge (i, j), (i, k) coectig these rectagles, let us deote the smallest label occurrig i (i, j) by η i,j Let η( (i, j), (i, k) ) := 3 3 /8 + (i 1)(/ + i)/4 + (k /) + (j 1)/ To see that this part of the labellig is well-defied, three observatios are eeded for /4 < i /, 1 j / < k : η i,j+1 η i,j > /4, η i+1,i η i,/ /4 + /4 η i+1,j η i,j > /4 To label the edges of the rectagle T (), we use up the leftover 3 /64 umbers of [7 3 /64 + 1, 3 3 /64] as a simple block This completes the defiitio of the fuctio η, as by the symmetry it was eough to defie the labellig up to 3 / We eed to show that the badwidth of the labellig η is ideed less tha 3 3 / /8: The largest differeces betwee the labels of two edges e ad f havig commo edpoits, up to the cetral symmetry, are cotaied i the followig list: (i) The edge e is betwee rectagles T (1) ad T (), ad f is i T () By defiitio, η(e) ( a)(3/ a)/4 ad η(f) /, implyig that η(f) η(e) < 3 3 /8 (ii) The edge e is iside of T (3) ad f is betwee T (3) ad T (6), havig there commo edpoit i the t-th row The η(e) (t 1)(/ + t 3)/4 ad η(f) 3 3 /64 + t /4 These bouds ad a optimizatio i the variable t shows that η(f) η(e) 3 3 /64 + t /8 t /4 + /8 + t 3/4 3 3 /8 + / (iii) The edge e is i the rectagle T (4) ad f is betwee T (4) ad T (5), with their commo edpoit i the t-th row, where /4 < t / The η(e) (t 1)(/ + t 3)/4 ad η(f) (t 1)(/ + t)/ /8 + / + (/ 1)/ It is easy to check that η(f) η(e) 3(t 1)/ /8 + / + /4 / 3 3 /8 + 7 /8 (iv) The edge e is betwee T (3) ad T (), ad f is betwee T () ad T (1), with their commo edpoit i the t-th row, where 1 t a The η(e) 3 3 /64 ad η(f) 3 ( a)(3/ a)/4 < 3 7/8 11/8 (1/4) = /56 implyig η(f) η(e) 3 3 /8 (v) The edge e is betwee T (3) T (4) ad T (1), ad f is i T (1), with their commo edpoit i the t-th row, where a < t / The η(e) (t 1)(/+t)/4+3 3 /8 ad η(f) 3 ( t 1)(/ + t 3)/4, implyig η(f) η(e) 3 /4 + t / t / + 19 /8 3 3 / /8 Proof of the lower boud Fix a labellig η Cosider S := S ( 1)/8(η), defied as the set of edges receivig labels from [ ( 1)/8] Let C deote the collectio of colums ad R the collectio of rows, cotaiig a edpoit of a edge from S We shall give a lower boud o the cardiality of the -shadow of S:

11 EDGE BANDWIDTH 11 A edge of S is determied by its two edpoits The first ca be chose from the set R C, the secod either from the leftover rows or colums, that is R + C ways Sice we have couted all edges twice ad S = ( 1)/8, we have 1 R C ( C + R ) ( 1) = S 8 This yields ( R + C ) ( C + R ) ( 1) by the arithmetic-geometric meas iequality, which implies that C + R >, or C + R + 1, because of the itegrality of the left had side A similar coutig argumet gives a lower boud o the -shadow σ (S) From the set of all edges of K K, we leave out the set S ad those edges havig both edpoits outside of C ad R (6) σ (S) ( 1) ( 1) 8 ( ) ( ) R C ( C ) ( R ) Note, that also by the arithmetic-geometric meas iequality ( ) ( ) R C ( C ) + ( R ) 1 ( C R ) ( C R ) Sice C + R + 1, we also have 1 ( C R ) ( ( 1) ( 3) C R ) 8 Developig (6), ad pluggig i the iequalities above, oe gets that is by Propositio 6 σ (S) , B (K K ) σ (S) The hypercube I this sectio we shall prove Theorem 3 Let us start with the upper boud First we eed the followig techical estimate Lemma 8 Let k be two itegers, ad fix 1 i 1 < < i k itegers The ( ) ij ( ) ( ) ij (7) ( k) ( k 1) = o() k + 1 j k + j /

12 1 BALOGH, MUBAYI, AND PLUHÁR Proof First we rewrite the left had side of (7): ( k) ( k 1) ( ) ij ( k 1) k + 1 j { ( ) ( ) i j ij } + k + 1 j k + j ( ) ij = k + j ( ) ij k + 1 j We eed some case aalysis to hadle the terms ( i j ) ( k+1 j ij k+ j), j = 1, k First assume that i j 3 log The ( ij k + 1 j ) < i j < 3 < 1 ( ) This meas that i this case these terms cotribute very little to the total sum Now we ca assume that j i j < 3 log We shall use the followig idetity: ( ) ( ) a a (8) = b + 1 a ( ) a = b + 1 a ( ) a + 1 b b + 1 b + 1 b a + 1 b + 1 If k < / log the usig (8) we obtai { ( ) ( ) i j ij } ( k 1) = ( k 1) k + j + i ( ) j ij < k + 1 j k + j k + j k + 1 j ( ) ij k + 3 log ( ) ( ) ( k 1) k + 1 j k + 3 log < O log There are at most 3 log of these terms, so their cotributio to the fial sum is egligible If k / log, we ca use the followig iequalities for t = /( log ), ( / +t ( / ) ) = ( / t + 1) ( / ) ( / + 1) ( / + t) ( t ) t 1 < exp( /( log 3 )) < 1 / + t 3 < ( / ) t = / + t Now we give a labellig of the edges of P with badwidth ( + o())( / ) We ca associate a set A x {1,, } to ay vertex x of P such that i A x iff the ith coordiate of x is 1 A edge ca be idetified with its two edpoits as (A, A + e), where A is a subset, ad e {1,, } A The labellig is doe by a variat of lexicographic order, defied as follows The first edge gets label 1, the secod gets label ad so o The order is: (A, A + e) < (B, B + f)

13 iff oe of these three coditios holds: A < B A = B ad mi{a B} A A = B ad e < f EDGE BANDWIDTH 13 Remark Ideed, this is othig else tha a appropriate breadth-first search labellig of the edges of the dimesioal cube startig from the origi The followig picture will show the details of this procedure for the three dimesioal cube {3} {, 3} {1,, 3} {1, 3} 3 {} {1, } {1} Labellig of P 3 Figure 3 I order to estimate the differeces arisig i meetig edges, we have to check the three differet possibilities for the edges to meet (i) If the two edges are of type (A, A + e) ad (A, A + f), the clearly there are at most 1 edges betwee them, hece the differece of their labels is at most (ii) Suppose edges of the form (A + e, A + e + f) ad (A + f, A + e + f) meet Without loss of geerality we may assume that e < f Let us estimate the umber of edges of the form (B, B + g), such that (A + e, A + e + f) < (B, B + g) < (A + f, A + e + f) The coditios above mea that A + e B A + f, from which A + e = B = A + f For fixed A, e, f, the set B ca be chose at most ( B ) ways, ad whe B is fixed, g ca be chose at most B ways Altogether the differece of the labels of (A + e, A + e + f) ad (A + f, A + e + f) is at most ( ) ( ) ( B ) B 6

14 14 BALOGH, MUBAYI, AND PLUHÁR (iii) Fially we cosider edges of the form (A, A + e) ad (A + e, A + e + f) Agai, we eed to estimate the umber of edges betwee these two edges i the give order Let (A, A + e) < (B, B + h) < (A + e, A + e + f), where A = {i 1,, i k } Observe that the vertex A + e has k + 1 eighbors of size k, ad the first oe amog them i our orderig is {A+e} mi{a+e} Hece the differece of the labels of the vertices A + e ad A is the largest (ie the umber of edges (B, B + h) betwee (A, A + e) ad (A + e, A + e + f) satisfyig B = k is maximized) whe e is maximal possible, therefore we may assume i k < e If B = k, the mi{a B} A ad the umber of such edges is ( ) ij ( k) k + 1 j If B = k + 1, the mi{{a + e} B} B ad the umber of such edges is ( ) { ( ) ( ) ij e } ( k 1) ( k 1) + k + 1 k + j e Cosequetly, the umber of edges betwee (A, A + e) ad (A, A + f) is ( ) ( k 1) +( k) k + 1 ( ) ij ( k 1) k + 1 j Now, by Lemma 8, ( ) ij ( k) ( k 1) k + 1 j { ad by the iequality (which could be easily checked) ( ) ( ) ( k 1) (/) k + 1 / ( ) ( ) ij e } + k + j e ( ) ( ) ij = o() k + j / the upper boud is proved The proof of the other directio is a refiemet of the proof of Calamoeri, Massii ad Vrťo [3] As we shall follow their proof ad otatio also, first we give the sketch of their ideas, too They picked a arbitrary edge set S of size ( from E(P ), ad showed that either (S) or (E(P ) S) is at least of size 4 / ), the applied Propositio 4 (Note that ( = E(P ) /) We make a more subtle case aalysis: if say (S) is greater tha / ), the we are doe by Propositio 4, while i the other case we take (S), 3 (S) ad so o, ad use Propositio 6 So let us fix a edge labellig of P Let S deote the set of edges labelled by {1,,, }, ad color the edges i S by red, ad rest of the edges by white For

15 EDGE BANDWIDTH 15 a vertex x V (P ), let E(x) deote the set of edges icidet to x Call a vertex x red if every edge i E(x) is red, white if every edge i E(x) is white, ad the rest is mixed Let R, W ad M deote the set of red, white ad mixed vertices, respectively Certaily, (9) R + W + M = For x M, let r(x) deote the umber of red edges i E(x), that is 1 r(x) 1 Furthermore, by (double) coutig the red edges we have that (10) R + x M r(x) = 1 = W + x M( r(x)) From the defiitio of mixed vertices we ca coclude the followig two iequalities: 1 (11) ( r(x)) (S) ad 1 r(x) (E(P ) S) x M Combiig these two iequalities we obtai M 4 (S) + (E(P ) S) x M max{ (S), (E(P ) S) } If ( / ) M, the by Propositio 4 we prove the required lower boud From ow o we therefore assume ( ) (1) M < Either r(x) M / x M or ( r(x)) M /; let us assume that the first iequality holds, sice otherwise we could switch the role of the red ad white vertices Combiig this with (10) ad (1) we obtai that 1 R + 1 ( ) M < R +, implyig the lower boud (13) 1 x M ( ) < R < 1 Note that the upper boud o R i (13) follows from (10) Combiig (1) ad (13) we obtai a upper boud for R M The lower boud give below follows from W < 1 ( ) (14) 1 < R M 1 +

16 16 BALOGH, MUBAYI, AND PLUHÁR We eed the followig easy observatio o the l-shadows: (15) x σ l (R M)E(x) σ l+1 (S) To estimate x σ l (R M) E(x) we eed a classical result of Harper [6] (see also a proof of it by Katoa [11]) Lemma 9 Let A V (P ), 0 y < ad r be a iteger such that 1 ( ) ( ) ( ) ( ) y A = r + 1 r The (A) ( ) ( ) y + r r 1 ( ) ( ) y r r 1 First, for some 0 y 0 < ad a iteger r 0 we have ( ) ( ) ( ) R M = r By (14) we have that / 4 r 0 / We shall apply Lemma 9 first to the set R M the repeatedly to + ( y0 R M σ(r M), R M σ (R M),, R M σ l 1 (R M) for l = 1/3 4 To do so, for 1 t l 1 write ( ) ( ) ( ) (16) R M σ t (R M) = r t + 1 where y t ad r t is a iteger By Lemma 9, (17) σ ( R M σ t (R M) ) ( ) r t 1 Note that the sequece {r t } is mootoe decreasig, ad as r / it meas that the sequece ( r t 1) is mootoe decreasig also If rl 1 / l 3 the by (17) l 1 σ l (R M) t=0 ( r t 1 ) ( l l 4 If r l < / l 3, the takig the differece of (14) ad (16) (these are disjoit sets) we have ( ) ( ) ( ) ( ) ( ) σ l (R M) + + > + + / 1 r l + 1 / / 4 r l Note that for a real y, ( y r) is defied as y (y 1) (y r + 1)/r! r 0 ) ) ( yt r t ),

17 EDGE BANDWIDTH 17 ad we ca coclude that ( ) (18) σ l (R M) l l 3 (Note that we assume that is large) That is by Propositio 6, (18) ad (15) we have B (P ) σl (R M) (l + 1) ( 1 1 l + 1 ) ( ) l 4 We estimate the rightmost expressio of the iequality above with l = 1/3 4 = t 4: ( ) / t ( / ) = ( / t + 1) ( / ) ( / + 1) ( / + t) ( t ) t 1 1 1/3 / + 1 This proves the lower boud of the theorem 6 Remarks ( / t + 1 ) t > / + 1 We believe that i Theorem 1 the upper bouds are the real values of the edgebadwidths Alas, it is hard eve to cojecture the exact value of B (K K ); we have o good cadidate for this O the other had, it is reasoable to thik that the upper boud, ad the labellig give i the proof of Theorem 3 is optimal Still, to show this will require more refied methods Fially, let T l be the graph whose vertices are the triples of o-egative itegers summig to l, with a edge coectig two triples if they agree i oe coordiate ad differ by 1 i the other two coordiates Hochberg, McDiarmid ad Saks [8] showed i a beautiful paper that B(T l ) = l + 1 It is atural to raise the followig questio Problem 10 What is the value of B (T l )? Ackowledgmet referees We would like to thak the useful advices of the ukow Refereces [1] J Balogh, J A Csirik, Idex assigmet for two-chael quatizatio, IEEE Tras Iform Theory 11 (004) [] S Bezrukov, N Grüwald ad K Weber, O edge umberigs of the -cube graph, Discrete Appl Math 46 (1993) [3] T Calamoeri, A Massii ad I Vrťo, New results o edge-badwidth, Theoret Comput Sci 307 (003)

18 18 BALOGH, MUBAYI, AND PLUHÁR [4] F R K Chug, Labeligs of graphs, i: L W Beieke, RJ Wilso (Eds) Selected topics i graph theory Vol 3, Academic Press, Sa Diego, CA, (1988) [5] J Chvátalová, Optimal labellig of a product of two paths Discrete Math 11 (1975) [6] L H Harper, Optimal umberigs ad isoperimetric problems o graphs, J Combiatorial Theory 1 (1966) [7] L H Harper, Global Methods for Combiatorial Isoperimetric Problems, 3 pp,cambridge Uiversity Press [8] R Hochberg, C McDiarmid ad M Saks, O the badwidth of triagulated triagles, Discrete Math 138 (1995) [9] F K Hwag ad J C Lagarias, Miimum rage sequeces of all k-subsets of a set, Discrete Math 19 (1977) [10] T Jiag, D Mubayi, A Shastri ad D B West, Edge-badwidth of graphs, SIAM J Discrete Math 1 (1999), [11] G O H Katoa, The Hammig-sphere has miimum boudary, Studia Sci Math Hugar 10 (1975) [1] Y L Lai ad K Williams, A survey of solved problems ad applicatios o badwidth, edgesum, ad profile of graphs, Joural of Graph Theory, 31 (1999), o, [13] Q Li, M Q Tao ad Y Q She, The badwidth of the discrete tori C m C, J Chia Uiv Sci Tech 11 (1981) 1 16 [14] C H Papadimitriou, The NP-completeess of the badwidth miimizatio problem, Computig 16 (1976) Departmet of Mathematical Scieces, The Ohio State Uiversity, Columbus, OH 4310, USA address: jobal@mathohio-stateedu Departmet of Mathematics, Statistics, ad Computer Sciece, Uiversity of Illiois, Chicago, IL 60607, USA address: mubayi@mathuicedu Uiversity of Szeged, Departmet of Computer Sciece, Árpád tér, Szeged, H-670 address: pluhar@ifu-szegedhu