LONG SNAKES IN POWERS OF THE COMPLETE GRAPH WITH AN ODD NUMBER OF VERTICES

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1 J Lodo Math Soc (2 50, (1994, LONG SNAKES IN POWERS OF THE COMPLETE GRAPH WITH AN ODD NUMBER OF VERTICES Jerzy Wojciechowski Abstract I [5] Abbott ad Katchalski ask if there exists a costat c > 0 such that for every d 2 there is a sake (cycle without chords of legth at least c3 d i the product of d copies of the complete graph K 3 We show that the aswer to the above questio is positive, ad that i geeral for ay odd iteger there is a costat c such that for every d 2 there is a sake of legth at least c d i the product of d copies of the complete graph K 1991 Mathematics Subject Classificatio 05C35, 05C38 1

2 1 Itroductio By a path i a graph G we mea a sequece of distict vertices of G with every pair of cosecutive vertices beig adjacet A path will be called closed if its first vertex is adjacet to the last oe By a chord of a path P i a graph G we mea a edge of G joiig two ocosecutive vertices of P If e is a chord i a closed path P, the e is called proper if it is ot the edge joiig the first vertex of P to its last vertex Note that a proper chord of a closed path correspods to the stadard otio of a chord i a cycle A sake i a graph G is a closed path i G without proper chords, ad a ope sake i G is a path i G without chords If G ad H are graphs, the the product of G ad H is the graph G H with V (G V (H as the vertex set ad (g 1, h 1 adjacet to (g 2, h 2 if either g 1 g 2 E(G ad h 1 = h 2, or else g 1 = g 2 ad h 1 h 2 E(H Let K d be the product of d copies of the complete graph K, 2, d 1 It will be coveiet to thik of the vertices of K d as d-tuples of -ary digits, ie the elemets of the set {0, 1,, 1}, with edges betwee two d-tuples differig at exactly oe coordiate Let S(K d be the legth of the logest sake i K d The problem of estimatig the value of S(K2 d (kow also as the sake-i-the-box problem has a extesive literature See [2], [6], [8], ad the refereces i these papers Evdokimov [6] was the first to prove that for some costat c > 0, S(K2 d c2 d (1 The costat c he got i his proof is very small (c = 2 11 Other shorter proofs of (1 ad with larger values of the costat c have bee give by Abbott ad Katchalski [2] ad Wojciechowski [8] The best lower boud for S(K d 2 has bee give by Abbott ad Katchalski [4] They show that (1 holds with c = = I [9] Wojciechowski proves a result of a differet but related ature, amely, that for every d 2, K d 2 ca be decomposed ito 16 vertex disjoit sakes 2

3 Whe 3, the existece of sakes of legth c d, where c is a positive costat idepedet of d, is kow oly for eve Abbott ad Katchalski [5] proved that if 0 mod 4, the for d 3 ( S(K d 2 d 1 S(K d 1 2 (2 It follows from (1 ad (2 that i the case whe 0 mod 4 the followig theorem is true Theorem 1 For ay eve iteger 2, there exists a costat c > 0 such that S(K d c d, (3 for every d 2 Whe 2 mod 4, the validity of Theorem 1 follows from the remark i [5] that the costructio used i the proof of (2 ca be modified to give log sakes i the case whe 2 mod 4 For odd, the best kow lower bouds o S(K d are ot as good as for eve Abbot ad Dierker [1] proved that d 1 d 2 k S(K d for 2, d 2, 2 2 k=1 ad { S(K d 2 d l 1 if 2 l < d < 2 l+1 2 d l if d = 2 l It follows that for every 2 there is a costat λ > 0 such that S(K d 1 d λ d Abbott ad Katchalski [5] ask whether there exists a costat c 3 > 0 such that (3 holds for = 3 I this paper we give a positive aswer to this questio, ad what is more, we show that the followig geeral result is true 3

4 Theorem 2 For ay odd iteger 3, there exists a costat c > 0 such that S(K d c d, for every d 2 Actually, we prove a stroger result Theorem 3 For ay odd iteger 3, ad ay d 5 S(K d 2( 1 d 4 Thus, i particular, for = 3 we get S(K d d The proof of Theorem 3 will be give i sectio 4 2 Basic Defiitios A m-path i a graph is a path cotaiig m vertices, ie a path of legth m 1 If P is a m-path, the we will write m = P A chai C of paths i a graph G is a sequece (P 1, P 2,, P m of paths i G such that each path i C has at least two vertices, ad the last vertex of P i is equal to the first vertex of P i+1, i = 1, 2,, m 1 Whe the umber m of paths i a chai eeds to be specified, we shall refer to a m-chai of paths A m-chai C = (P i m i=1 of paths will be called closed if the first vertex of P 1 is equal to the last vertex of P m Now we are goig to defie a importat operatio that will be used throughout the paper Give a m-path P = (g i m i=1 i a graph G ad a m-chai of paths C = (P i m i=1 i a graph H, let P C be the ( m i=1 P i -path i the graph G H costructed i the followig way For each path P i = (h i1, h i2,, h iki i C let P i be the path 4

5 ( (gi, h i1, (g i, h i2,, (g i, h iki i G H Note that for each i {1, 2,, m 1} the last vertex of the path P i is adjacet to the first vertex of the path P i+1 Let P C be the path obtaied by joiig together (juxtaposig the paths P 1, P 2,, P m We will say that P C is the path geerated by P ad C Note that the path geerated by a closed path ad a closed chai of paths is a closed path If D is a km-chai of paths i a graph H, the the m-splittig of D is the sequece (D 1, D 2,, D m of k-chais of paths i H which joied together (juxtaposed give D The above defiitio of the operatio ca be geeralized to the followig situatio Let C = (P i m i=1 be a m-chai of k-paths i a graph G, let D be a km-chai of paths i H, ad let (D 1, D 2,, D m be the m-splittig of D Note that for each i {1, 2,, m 1} the last vertex of the path P i D i i the graph G H is equal to the first vertex of the path P i+1 D i+1 Set C D = (P 1 D 1, P 2 D 2,, P m D m We will say that C D is the chai of paths geerated by C ad D Note that the chai of paths geerated by two closed chais of paths is also a closed chai of paths It is straightforward to verify that the operatio is associative i the followig sese Property 1 If P is a m-path i a graph G, C is a m-chai of k-paths i a graph H, ad D is a km-chai of paths i a graph K, the ( P C D = P ( C D Let C = (P i m i=1 be a chai of paths i a graph G We say that C is opely separated if for i m 1 ad j = i + 1, P i ad P j have exactly oe vertex i commo, ad otherwise P i ad P j are vertex disjoit We say that C is closely separated if C is closed, P i ad P j have exactly oe vertex i commo whe either i m 1 ad j = i + 1, or i = 1 ad j = m, 5

6 ad otherwise P i ad P j are vertex disjoit The followig property is a straightforward cosequece of the defiitios ad its proof is left to the reader as a exercise Property 2 (i The path P C geerated by a path P i a graph G ad a opely separated chai C of ope sakes i a graph H is a ope sake i the graph G H (ii The closed path P C geerated by a closed path P i a graph G ad a closely separated chai C of ope sakes i a graph H is a sake i the graph G H If P is a path, the let P be the path obtaied from P by reversig the order of vertices, ad if C = (P i m i=1 is a chai of paths, the let C = ( P m, P m 1,, P 1 be the chai of paths obtaied from C by reversig the order of paths ad reversig every path The expressio ( 1 i X, where X is a path or a chai of paths, will mea X for i eve ad X for i odd Obviously, the followig property is true Property 3 If P is a m-path i a graph G ad C is a m-chai of paths i a graph H, the ( P C = (P C Let C be a km-chai of paths, ad let S = ( C 1, C 2,, C m be the m-splittig of C By the alterate matrix of the splittig S we mea the followig (m k-matrix A of paths: A = C 1 C 2 ( 1 m 1 C m = Q 1 1 Q 2 1 Q k 1 Q 1 2 Q 2 2 Q k 2 Q 1 m Q 2 m Q k m where (Q 1 i, Q2 i,, Qk i is the sequece of paths formig the k-chai ( 1i 1 C i The splittig S will be called opely alteratig if for every odd j, 1 j m 1, the paths Q k j ad Qk j+1 have exactly oe vertex i commo, for every eve j, 2 j m 1, the paths Q 1 j ad Q1 j+1 have exactly oe vertex i commo, ad otherwise the paths Q i j ad Qi l are vertex disjoit, i {1, 2,, k}, j, l {1, 2, m}, j l Note that the splittig S is opely alteratig 6

7 if for every colum of its alterate matrix A the paths i the colum are mutually vertex disjoit except for the shared vertices which are ecessary for C to be a chai of paths, ie Q k 1 ad Q k 2 have exactly oe vertex i commo, Q 1 2 ad Q 1 3 have exactly oe vertex i commo, ad so o Assume ow that the km-chai C is a closed chai of paths ad m is eve The, we say that the splittig S is closely alteratig if for every odd j, 1 j m 1, the paths Q k j ad Q k j+1 have exactly oe vertex i commo, for every eve j, 2 j m 1, the paths Q 1 j ad Q1 j+1 have exactly oe vertex i commo, the paths Q1 1 ad Q 1 m have exactly oe vertex i commo, ad otherwise the paths Q i j ad Qi l are vertex disjoit, i {1, 2,, k}, j, l {1, 2, m}, j l As above, ote that the splittig S is closely alteratig if for every colum of its alterate matrix A the paths i the colum are mutually vertex disjoit except for the shared vertices which are ecessary for C to be a closed chai of paths The followig property is a straightforward cosequece of the defiitios Property 4 Let P be a k-path i a graph G, ad let D be a km-chai of paths i a graph H (i If C is the m-chai (P, P,, ( 1 m 1 P, ad the m-splittig of D is opely alteratig, the the m-chai C D of paths i the graph G H is opely separated (ii If m is eve, C is the closed m-chai (P, P, P, P,, P, ad the m-splittig of D is closely alteratig, the the closed m-chai C D of paths i the graph G H is closely separated Assume that 3 is a fixed odd iteger For every iteger d 1, we are goig ow to defie the d -path π d i K, d ad the closed ( 1 d -path γ d+1 i K d+1 These paths will be used i the costructio of log sakes Let π 1 be the -path (0, 1,, 1 i K, ad let γ be the closed ( 1-path 7

8 (0, 1,, 2 i K If d 1 ad the path π d i K d is defied, the let π d+1 = π 1 ( π, d π, d π, d π, d, π d ad γ d+1 = γ ( π d, π d, π d, π d,, π d Let H be a graph, d 1 be a iteger, C be a d -chai of paths i H, ad D be a ( 1 d -chai of paths i H We say that C is opely well distributed if either d = 1 ad C is a opely separated chai of ope sakes, or d 2, every chai C i i the -splittig S = (C 1, C 2,, C of C is opely well distributed ad S is opely alteratig We also say that D is closely well distributed if every chai D i i the ( 1-splittig S = (D 1, D 2,, D 1 of D is opely well distributed ad S is closely alteratig The followig property ca be proved by a straightforward iductio with respect to d Property 5 If C is a opely well distributed d -chai of paths i a graph H, the the chai C is also opely well distributed Now we are ready to prove the followig importat lemma Lemma 1 If d 1 ad C is a opely well distributed d -chai of paths i a graph H, the the path π d C is a ope sake i the graph K d H Proof We are goig to use iductio with respect to d For d = 1, the lemma is true by Property 2 (i Assume that d 1, ad that C is a opely well distributed d+1 -chai of paths i the graph H Let S = (C 1, C 2,, C be the -splittig of C By the defiitio, we have π d+1 π d+1 C = (π 1 D C, where D = (π d, π d,, π d By Property 1, the chai C is equal to π 1 (D C We have D C = (π d C 1, ( π d C 2,, π d C 8

9 By Property 3, D C = (π d C 1, (π d C 2,, π d C Sice the chai C is opely well distributed, the chais C 1, C 2,, C are also opely well distributed By Property 5, the chais C 1, C 2,, C are opely well distributed, so by the iductive hypothesis the paths π d C 1, π d C 2,, π d C are ope sakes i K d H Hece D C is a chai of ope sakes i K d H The splittig S is opely alteratig, so by Property 4 (i, the chai D C is opely separated Hece by Property 2 (i, π 1 (D C is a ope sake i K K d H = K d+1 H, ad the proof is complete The followig lemma will be used i the proof of Theorem 3 Lemma 2 If d 1 ad C is a closely well distributed ( 1 d -chai of paths i a graph H, the the path γ d+1 C is a sake i the graph K d+1 H Proof Assume that d 1 ad that C is a closely well distributed ( 1 d -chai of paths i the graph H Let S = (C 1, C 2,, C 1 be the ( 1-splittig of C By the defiitio we have γ d+1 Property 1, the chai γ d+1 C = (γ D C, where D is the ( 1-chai (π d, π d,, π d By C is equal to γ (D C We have D C = (π d C 1, ( π d C 2,, ( π d C 1 By Property 3, D C = ( π d C 1, (π d C 2,, (π d C 1 By Property 5 ad Lemma 1, D C is a chai of ope sakes i K d The splittig S is closely alteratig so by Property 4 (ii, the chai D C is closely separated Hece by Property 2 (ii, γ (D C is a sake i K K d H = K d+1 H, ad the proof is complete 9

10 3 Costructio of Well Distributed Chais Let C be a chai of paths i a graph G We say that C jois u 1 to u 2 if u 1 is the first vertex of the first path of C ad u 2 is the last vertex of the last path If v is the first or the last vertex of a path P, the let P v be the path obtaied by removig v from P Let C = (P i m i=1 be a chai of paths joiig u 1 to u 2, ad D = (Q i m i=1 be a chai of paths joiig v 1 to v 2 We say that C ad D are parallel if for every i, 1 i m, the paths P i ad Q i are vertex disjoit We also say that C ad D are iterally parallel if the followig coditios are satisfied: (i the paths P 1 u 1 ad Q 1 are vertex disjoit, (ii the paths P 1 ad Q 1 v 1 are vertex disjoit, (iii the paths P m u 2 ad Q m are vertex disjoit, (iv the paths P m ad Q m v 2 are vertex disjoit, ad (v for every i, 1 < i < m, the paths P i ad Q i are vertex disjoit Let 2 be a iteger ad let H be a graph A -et i H is a pair (U, M, where U = {u 1, u 2,, u +3 } is a set of + 3 vertices of H ad M = { } Ns,t i : i {0, 1,, 1}; s, t {1, 2,, + 3}; s t is a set of opely separated chais of ope sakes i H such that N i s,t jois u s to u t, N i s,t = N i t,s, ad if i j the N i s,t ad N j v,w are iterally parallel, for ay i, j {0, 1,, 1}, ad s, t, v, w {1, 2,, + 3}, s t, v w Assume ow that 3 is a fixed odd iteger Let X be the followig (+3-matrix X =

11 Let Y be the followig ( 1 ( + 3-matrix Y = The matrix X ca be obtaied by takig as rows the cosecutive cyclic permutatios of the sequece (3, 4,, + 3, 1, 2, ad the for each odd row exchagig the etries at the positios 1 ad 3 ad exchagig the etries at the positios 2 ad 4 The matrix Y ca be obtaied from X by removig the last row ad modifyig the 1 row From ow o, to the ed of this sectio, let us assume that we are give a graph H, ad a -et (U, M i H Let C be a d -chai of paths i H, d 1 If every -chai i the d 1 -splittig of C belogs to M, the we say that C is M-built For each i = 1, 2,,, let ϕ i : M M be defied by ϕ i (Nv,w j j+i 1 mod = Nx i,v,x i,w, where the bottom idices are take from the matrix X = (x p,r Aalogously, for each i = 1, 2,, 1, let ψ i : M M be defied by ψ i (Nv,w j j+i 1 mod = Ny i,v,y i,w, where the bottom idices are take from the matrix Y = (y p,r If C is ay M-built d - chai, the ϕ i (C ad ψ i (C are the d -chais obtaied by applyig ϕ i ad ψ i, respectively, to each of the -chais i the d 1 -splittig of C Let D 1 ad D 2 be M-built d -chais with d 1 -splittigs (N i k s k,t k d 1 k=1 ad (N j k v k,w k d 1 k=1 respectively If i k = j k for 1 k d 1, s k = v k for 1 < k d 1, ad t k = w k for 1 k < d 1, the we say that D 1 ad D 2 are iterally M-compatible If i k j k for 1 k d 1, s k v k for 1 < k d 1, ad t k w k for 1 k < d 1, the we say that 11

12 D 1 ad D 2 are iterally M-parallel If i k j k for 1 k d 1, s k v k for 1 k d 1, ad t k w k for 1 k d 1, the we say that D 1 ad D 2 are M-parallel Let C be a M-built d+1 -chai, let S = (C 1, C 2,, C be the -splittig of C, ad let A = C 1 C 2 C 3 C 1 C be the alterate matrix of S We say that C is M-well distributed if either d = 0, or d > 0 ad the followig coditios are satisfied (i For every i, 1 i, the d -chai C i is M-well distributed (ii Ay two cosecutive rows of A form a pair of iterally M-parallel chais (iii Ay two ocosecutive rows of A form a pair of M-parallel chais Sice the -splittig of a M-well distributed chai is opely alteratig, the followig property ca be proved by iductio with respect to d Property 6 If C is a M-well distributed d -chai, the C is a opely well distributed chai Sice o iteger appears twice i ay row of X, o iteger appears twice i ay row of Y, ad N i s,t = N i t,s, a straightforward iductio with respect to d ca be used to prove the followig property Property 7 If C is a M-well distributed d -chai, the for ay i {1, 2,, } ad ay j {1, 2,, 1} the chais ϕ i (C, ψ j (C, ad C are also M-well distributed Now we are ready to prove the followig importat lemma Lemma 3 For ay d 1 there exist four iterally M-compatible d -chais C d 3,1, C d 3,2, C d 4,1 12

13 ad C d 4,2, such that C d s,t is M-well distributed ad jois u s to u t, for ay s {3, 4}, t {1, 2} Proof We shall use iductio with respect to d If d = 1, the let C 1 s,t = N 0 s,t, for each s {3, 4}, ad each t {1, 2} Clearly, the -chais C 1 3,1, C 1 3,2, C 1 4,1 ad C 1 4,2 have the required properties Suppose ow that d 1 ad the d -chais C d 3,1, C d 3,2, C d 4,1 ad C d 4,2 satisfy the specified coditios Give s {3, 4} ad t {1, 2}, let C d+1 s,t be the d+1 -chai with the -splittig S = (C 1, C 2,, C such that C 1 = ϕ 1 (Cs,1, d C i = ϕ i (C4,1 d for odd i, C i = ϕ i (C3,2 d for eve i, 1 < i <, ad C = ϕ (C4,t d Let m = d 1 Sice C3,1, d C3,2, d C4,1, d ad C4,2 d are iterally M-compatible, there are i 1, i 2,, i m {0, 1,, 1} ad z 1, z 2,, z m 1 {1, 2,, + 3} such that for ay v {3, 4} ad w {1, 2}, the sequece (N i 1 v,z 1, N i 2, N i 3 z 2,z 3,, N i m 1 z m 2,z m 1, N i m zm 1,w is the m-splittig of C d v,w Thus, if A is the alterate matrix of the splittig S, the we have: C 1 C 2 C 3 C A = 4 C 5 C 1 C = ϕ 1 (C d s,1 ϕ 2 (C d 3,2 ϕ 3 (C d 4,1 ϕ 4 (C d 3,2 ϕ 5 (C d 4,1 ϕ 1 (C3,2 d ϕ (C4,t d = ϕ 1 (N i 1 s,z 1 ϕ 1 (N i 2 ϕ 1 (N i m zm 1,1 ϕ 2 (N i 1 3,z 1 ϕ 2 (N i 2 ϕ 2 (N i m zm 1,2 ϕ 3 (N i 1 4,z 1 ϕ 3 (N i 2 ϕ 3 (N i m zm 1,1 ϕ 4 (N i 1 3,z 1 ϕ 4 (N i 2 ϕ 4 (N i m zm 1,2 ϕ 5 (N i 1 4,z 1 ϕ 5 (N i 2 ϕ 5 (N i m zm 1,1 ϕ 1 (N i 1 3,z 1 ϕ 1 (N i 2 ϕ 1 (N i m zm 1,2 ϕ (N i 1 4,z 1 ϕ (N i 2 ϕ (N i m zm 1,t We will prove ow that Cs,t d+1 satisfies the required coditios To verify that Cs,t d+1 a chai of paths, ote that the etries of the matrix X = (x j,k satisfy x i,1 = x i+1,2 for i odd, ad x i,3 = x i+1,4 for i eve, 1 i 1 Sice x 1,s = s ad x,t = t, the chai C d+1 s,t jois u s to u t Sice the chais C d v,w are M-well distributed, Property 7 implies that the chais C 1, C 2,, C are also M-well distributed Sice the itegers 3 ad 4 do ot appear i the 3rd or 4th colum of the matrix X except i the first row, the itegers 1 ad 2 do 13 is

14 ot appear i the 1st or 2d colum of X except i the last row, ad o iteger appears twice i ay colum of X, it follows from the defiitio of the fuctios ϕ i that ay two cosecutive rows of A are iterally M-parallel chais ad ay two ocosecutive rows of A are M-parallel chais Hece the chai C d+1 s,t is M-well distributed Sice it is clear that the chais C d+1 3,1, Cd+1 3,2, Cd+1 4,1 ad C d+1 4,2 are iterally M-compatible, the proof is complete The followig lemma will be used i the proof of Theorem 3 Lemma 4 For ay d 1 there exists a closely well distributed M-built ( 1 d -chai Proof Let d 1 Let D d+1 be the closed ( 1 d -chai with the ( 1-splittig S = (C 1, C 2,, C 1 such that C i = ψ i (C d 4,1 for odd i, ad C i = ψ i (C d 3,2 for eve i, 1 i 1 Thus, if A is the alterate matrix of the splittig S, ad m = d 1, the we have: A = C 1 C 2 C 3 C 2 C 1 = ψ 1 (C d 4,1 ψ 2 (C d 3,2 ψ 3 (C d 4,1 ψ 2 (C4,1 d ψ 1 (C3,2 d = ψ 1 (N i 1 4,z 1 ψ 1 (N i 2 ψ 1 (N i m zm 1,1 ψ 2 (N i 1 3,z 1 ψ 2 (N i 2 ψ 2 (N i m zm 1,2 ψ 3 (N i 1 4,z 1 ψ 3 (N i 2 ψ 3 (N i m zm 1,1 ψ 2 (N i 1 4,z 1 ψ 2 (N i 2 ψ 2 (N i m zm 1,1 ψ 1 (N i 1 3,z 1 ψ 1 (N i 2 ψ 1 (N i m zm 1,2 where, for some i 1, i 2,, i m {0, 1,, 1}, ad z 1, z 2,, z m 1 {1, 2,, + 3}, the sequece (N i 1 4,z 1, N i 2, N i 3 z 2,z 3,, N i m 1 z m 2,z m 1, N i m zm 1,1, is the m-splittig of the chai C d 4,1, ad the sequece (N i 1 3,z 1, N i 2, N i 3 z 2,z 3,, N i m 1 z m 2,z m 1, N i m zm 1,2 is the splittig of the chai C d 3,2, Sice the etries of the matrix Y = (y j,k satisfy y i,1 = y i+1,2 for i odd, y i,3 = y i+1,4 for i eve, 1 i 2, ad y 1,3 = y 1,4, it follows that the sequece D d+1 of paths is a closed chai of paths Sice the chais C d v,w are M-well distributed, Property 7 implies that 14

15 the chais C 1, C 2,, C 1 are also M-well distributed By Property 6, C 1, C 2,, C 1 are opely well distributed Sice o iteger appears twice i ay colums of Y, it follows that ay two cosecutive rows of A are iterally M-parallel chais ad ay two ocosecutive rows of A are M-parallel chais, hece the splittig S is closely alteratig Thus, D d+1 a closely well distributed M-built chai, ad the proof is complete is 4 Costructio of a -Net i K 3 Let 3 be a fixed odd iteger The vertices of the graph K 3 are 3-tuples of digits from the set {0, 1,, 1} Give a vertex v = (a 1, a 2, a 3 of K 3, we say that a i appears at the i-th positio of v, i = 1, 2, 3 If a {0, 1,, 1}, the let a = a mod, ad a = a mod If a 1, a 2 {0, 1,, 1}, the let [a 1, a 2 ] = {a 1, a 1 +1, a 1 +2,, a 2 }, where the additio is take modulo Thus, for example, if = 7 the [5, 1] = {5, 6, 0, 1} We are goig ow to defie the -et (U, M i K 3 Let { U = {u 1, u 2,, u +3 } = (a 1, a 2, a 3 : a 1 {0, 1,, } }, {a 2, a 3 } = {a 1, a 1 } To costruct the chais i the set M we eed to itroduce some defiitios Suppose that v = (a 1, a 2, a 3 is a vertex from the set U, ad let us assume that the digit b {a 1, a 2, a 3 } The adjuct of the digit b i v is the digit a 1 if b [a 1, a 1 ], ad the digit a 1 otherwise Let η : {1, 2, 3} {1, 2, 3} be the fuctio defied by η(i = i + 1 if 1 i < 3, ad η(3 = 1 For ay i {0, 1,, 1} ad s {1, 2,, + 3}, let Q i s be the path startig with u s defied as follows If the digit i appears at the j-th positio i u s, the let Q i s = (u s, u s, where u s is obtaied from u s by replacig the digit at the η(j-th positio with the digit i If the digit i does ot appear i u s, the let the adjuct of i appear at the j-th positio i u s Let Q i s = (u s, u s, u s, where u s is obtaied from u s by replacig the digit at the j-th positio with the digit i, ad let u s be obtaied from u s by replacig the digit at the η(j-th positio with the digit i For example, if = 5 ad the sequece u 1, u 2,, u 8 of vertices i U 5 is 15

16 023, 032, 134, 143, 240, 204, 301, 310, where we write a 1 a 2 a 3 istead of (a 1, a 2, a 3, the the path Q i s is the path i the s-th row ad the (i + 1-st colum of the followig matrix: (023, 003 (023, 013, 011 (023, 022 (023, 323 (023, 024, 424 (032, 002 (032, 031, 131 (032, 232 (032, 033 (032, 042, 044 (134, 130, 030 (134, 114 (134, 124, 122 (134, 133 (134, 434 (143, 103, 100 (143, 113 (143, 142, 242 (143, 343 (143, 144 (240, 040 (240, 241, 141 (240, 220 (240, 230, 233 (240, 244 (204, 200 (204, 214, 211 (204, 224 (204, 203, 303 (204, 404 (301, 300 (301, 101 (301, 302, 202 (301, 331 (301, 341, 344 (310, 010 (310, 311 (310, 320, 322 (310, 330 (310, 314, 414 It ca be verified that the followig property is true Property 8 If i, j {0, 1,, 1}, i j, ad s, t {1, 2,, + 3}, s t, the the paths Q i s ad Q j s have oly the first vertex i commo, ad the paths Q i s ad Q j t are vertex disjoit Now we are ready to defie the set M = { } Ns,t i : i {0, 1,, 1}; s, t {1, 2,, + 3}; s t of -chais of paths i K 3 Give i {0, 1,, 1}, ad s, t {1, 2,, + 3} such that s < t let Ns,t i = (Q i s, P 1, P 2,, P 2, Q i t, ad Nt,s i = (Q i t, P 2, P 3,, P 1, Q i s, where P 1, P 2,, P 2 are defied as follows Let v s, v t be the last vertices of the paths Q i s ad Q i t respectively, let j s, j t {1, 2, 3} be such that i does ot appear either at the j s -th positio of v s, or at the j t -th positio of v t, ad let a s, a t be the digits at the positio j s of v s ad the positio j t of v t respectively Note that for v {v s, v t }, the digit i appears at exactly two positios of v, hece j s ad j t are well defied ad i {a s, a t } Now let us cosider two cases 16

17 If j s = j t, the let b 2, b 3,, b 2 be a sequece of differet digits from the set {0, 1,, 1} \ {a s, a t, i}, let b 1 = a s, b 1 = a t, ad let P j be the path (w j, w j+1 i K 3, j = 1, 2,, 2, where w k is obtaied from v s by replacig the digit at the i-th positio with the digit b k, k = 1, 2,, 1 If j s j t, the let m = 1 2, let b 1 = a s, let b 2, b 3,, b m be a sequece of differet digits from the set {0, 1,, 1} \ {a s, i}, let b m+1, b m+2,, b 2m 1 be a sequece of differet digits from the set {0, 1,, 1} \ {a t, i}, ad let b 2m = a t For j {1, 2,, 2m 1} \ {m} let P j be the path (w j, w j+1 i K 3, ad let P m = ( w m, (i, i, i, w m+1, where wk is obtaied from v s by replacig the digit at the j s -th positio with the digit b k, for k = 1, 2,, m, ad w k is obtaied from v t by replacig the digit at the j t -th positio with the digit b k, for k = m + 1, m + 2,, 2m Assummig that i the costructio of the paths P 1, P 2,, P 2, we choose the sequeces of digits to be icreasig ad cotaiig as small digits as possible, i the case = 5, with u 1, u 2,, u 8 as above, we have for example: ( N1,2 0 = (023, 003, (001, 001, (001, 004, (004, 002, (002, 032 ( N1,3 0 = (203, 003, (003, 001, (001, 000, 010, (010, 030, (030, 130, 134 ( N3,4 0 = (134, 130, 030, (030, 010, (010, 000, 200, (200, 100, (100, 103, 143 The followig lemma will be used i the proof of Theorem 3 Lemma 5 The pair (U, M is a -et i K 3 Proof Clearly, it follows from the costructio ad Property 8 that Ns,t i is a opely separated chai of ope sakes joiig u s to u t, ad Ns,t i = Nt,s, i for ay i {0, 1,, 1} ad s, t {1, 2,, + 3}, s t Note that for every path P of Ns,t i except the first ad the last, the digit i appears at least at two positios i each vertex of P From the above fact ad Property 8 it follows that Ns,t i ad Nv,w j are iterally parallel, for ay i, j {0, 1,, 1} ad s, t, v, w {1, 2,, +3}, such that i j ad v w, hece the proof is complete Now, we are ready to prove Theorem 3 17

18 Proof of Theorem 3 We have K d = K d 3 K 3 ad d 3 2 Let (U, M be the -et i K 3 defied i this sectio (see Lemma 5 By Lemma 4, there is a closely well distributed M -built ( 1 d 4 -chai D d 3 By Lemma 2, the closed path P = γ d 3 D d 3 is a sake i K d Sice ad each path i D d 3 γ d 3 = ( 1 d 4 has legth at least 2, we have S(K d P 2( 1d 4, ad the proof is complete 5 Cocludig remarks The costructio of log sakes i K d preseted i this paper ca, after mior modificatios, be used i the case whe is eve, ad thus we ca use it to prove Theorem 1 However, the value of the costat c obtaied i such a proof is ot as good as whe (1 ad (2 are applied (see Itroductio It follows from (2 ad (3 that there is a costat c > 0 idepedet of such that for ay 0 mod 4, ad ay d 2 S(K d c d 1 (4 By the remark i [5], the costat c ca be take i such a way that (4 holds for all eve A atural questio arises Questio 1 Is there a costat c > 0 such that for every 2 ad d 2 S(K d c d 1, We would like to ask a stroger questio 18

19 Questio 2 Is there a costat c > 0 ad a iteger d 0 such that for every 2 there is a -et (U d 0, M d 0 i the graph K d 0, where every chai C = (P i i=1 i the set Md 0 satisfies P i c d 0 i=1 It follows from the proof of Theorem 3 that the positive aswer to Questio 2 implies that the aswer to Questio 1 is also positive The best kow upper boud o S(K d 2 has bee give by Sevily [7] S(K d 2 2 d 1 2d 1 20d 41 I the geeral case, Abbott ad Katchalski [3] proved that S(K d ( d 1 d 1 Refereces [1] HL Abbott, PF Dierker, Sakes i powers of complete graphs, SIAM J Appl Math 32 (1977, [2] HL Abbott, M Katchalski, O the sake i the box problem, J Comb Theory Ser B 45 (1988, [3] HL Abbott, M Katchalski, Sakes ad pseudo sakes i powers of complete graphs, Discrete Math 68 (1988, 1 8 [4] HL Abbott, M Katchalski, O the costructio of sake i the box codes, Utilitas Mathematica 40 (1991, [5] HL Abbott, M Katchalski, Further results o sakes i powers of complete graphs, Discrete Math 91 (1991, [6] AA Evdokimov, The maximal legth of a chai i the uit -dimesioal cube Mat Zametki 6 (1969, Eglish traslatio i Math Notes 6 (1969, [7] HS Sevily, The sake-i-the-box problem: a ew upper boud, Discrete Math, to appear [8] J Wojciechowski, A ew lower boud for sake-i-the-box codes, Combiatorica 9 (1989,

20 [9] J Wojciechowski, Coverig the hypercube with a bouded umber of disjoit sakes, Combiatorica, to appear Departmet of Mathematics, West Virgiia Uiversity, PO BOX 6310, Morgatow, WV , USA UN020243@VAXAWVNETEDU JERZY@MATHWVUEDU 20