Some non-existence results on Leech trees

Transcription

1 Some o-existece results o Leech trees László A.Székely Hua Wag Yog Zhag Uiversity of South Carolia This paper is dedicated to the memory of Domiique de Cae, who itroduced LAS to Leech trees.. Abstract More tha 5 years ago Joh Leech [] published the followig beautiful problem: fid, wheever possible, trees o vertices with positive weights o the edges, such that the weighted distaces amog the vertices are exactly the umbers,, 3,...,. This paper makes a modest progress o this problem. Examples for Leech trees A tree is called Leech tree if oe ca assig positive edge weights to its edges, such that the path weights, i.e. the sums of weights alog the distict paths coectig the pairs of the vertices of the tree, yield exactly the umbers,, 3,...,. Sice edges of the tree are also paths, the edge weights have to be positive itegers as well. Joh Leech itroduced these trees i []. Believe it or ot, he was motivated by a problem of electrical egieerig, where edge weights represeted electrical resistaces. He gave a list of small Leech trees see Fig. ad posed the problem of their existece i geeral. The difficulty of the existece problem lies i the uusual way of mixig additive umber theory with combiatorics, i particular i the expoetial growth of the umber of cadidates for Leech The research of the author was supported i part by the NSF cotract DMS ad

2 trees. Leech wrote I expect the resolutio of this questio to be very difficult Figure : The kow Leech trees. Note that there are similar problems aroud that are otoriously hard. The Graceful Tree Cojecture of Rigel [3] states that for every tree with vertices, there is a bijectio f betwee the vertex set ad {,,..., } such that { fv fu : uv edge} = {,,..., }. The sevetee years old Prime Labelig Cojecture of Etriger states that for every tree with vertices, there is a bijectio f betwee the vertex set ad {,,..., } such that gcdfv,fu = wheever uv is a edge [4]. Results for the shape of Leech trees Herbert Taylor [6] gave a beautiful proof restrictig the umber of vertices o which Leech trees ca live. For completeess we also show his proof. Theorem. H. Taylor If there is a Leech tree o vertices, the = k or = k +. Proof. Let dx, y deote the sum of weights o the path coectig vertices x ad y. The crucial observatio is the fact that for ay 3 vertices x, y, z i a tree, oe has dx, y dx, z+dy, z mod.

3 Fix a vertex v ad let A deote the set of vertices lyig at a eve distace from v v is icluded, ad let B deote the set of vertices lyig at a odd distace from v. Nowwehave A + B =. Accordig to, two vertices defie a odd-legth path if ad oly if oe of them belogs to A, adthe other to B. Therefore the umber of paths with odd weight is A B.If is eve, there must be exactly paths with odd weights. Hece A B = A + B 4 A B = =. If is odd, there must be exactly Hece A B = A + B 4 A B = + paths with odd weights. + =. Note that the proof gave seemigly more tha the theorem: If there is a Leech tree o vertices, the = k if is eve, ad = k + if is odd. I fact, this more precise statemet gives o more restrictio. We leave this as a exercise to the Reader. Sice o examples of Leech trees for >6 are kow, ad i this paper we show further o-existece results, oe may arrive to the cojecture that Cojecture. There are oly fiitely may Leech trees. We give further support to this cojecture i the ext sectio, showig that there are o Leech trees o = 9 ad vertices. I the rest of this sectio we prove this cojecture for particular tree shapes. Leech oted without proof that a path ca be a Leech tree oly for 4. Ideed, sice must be amog the path weights, it has to be the weight of the full tree. All edges of the tree must have distict positive iteger weights, ad the i th smallest of them is at least i. Sice =, it follows that the edgeweightsmust be the umbers,,...,. Weight must be adjacet to ad othig else, otherwise weight ad the edge adjacet to it would yield a path of weight <, which is already preset as a edge. Sice ad already give a path of legth, ca be adjacet oly to, 3

4 otherwise a path with weight would be foud. Now we caot put ay weighted edge to the other side of. We ca geeralize this observatio by provig that Leech trees caot have very log paths: Theorem. IfthereisaLeechtreeovertices, the it has o paths loger tha + o. Proof. Assume that a,a,..., a t are the weights alog a path i a Leech tree o vertices i this order. Observe the followig iequalities: t a i i= t a i + a i+ 3 i= t j a i + a i a i+j j +. 4 i= Summig up these iequalities for j =0,,,..., k, we obtai that some t +t t k+=tk k distict itegers sum up to at most k+. Sice the i th smallest amog these distict itegers is, agai, at least i, we obtai that [ tk ][ k tk ] k + k+ Settig k = i 5, we obtai the theorem.. 5 We prove that largest Leech star cosists of edges with weights,, ad 4. Ideed, it is easy to see that ay Leech star with at least 3 leaves must cotai these edge weights. Assume that there is a fourth leaf. Its edge must have weight 7. However, the star with edge weights,,4, ad 7 is ot a Leech star, sice 0 does ot occur as a distace. If a Leech star with at least 5 leaves has edge weights,,4,7, ad 0, the is represeted twice as a distace, a cotradictio. The theorem below shows that Leech trees caot eve go close to the star shape. Theorem 3. I a Leech tree, the maximum degree of a vertex is at most d o. 4

5 Proof. Let us be give a Leech tree o vertices, which has a vertex v of degree d. Let W deote the set of d distict itegers, which are the weights of edges adjacet to v. Let D deote the set of d eighbors of v, ad let N deote the set of d distict distaces [ amog pairs of vertices i D. We will cosider the itervals I =, 3 ] I = 3, ],ad [ I 3 = 3, ]. Let i l deote W I l,forl=,,3. Let Z deote the set of d distaces which do ot have both edpoits i D. Let jl deote Z I l,forl=,,3. We have the followig iequalities: i i + j + j 3 j + j + j 3, 3 6, 3 7 d. 8 Formula 6 follows from the fact that the elemets of I are represeted as distaces. Formula 7 follows from the fact that the elemets of I 3 are represeted as distaces. Formulae 6, 7, ad 8 immediately imply that for l =, i l d +O. 9 3 Takig the squareroot of 9 for l =,, ad addig it up, oe obtais d i + i d + O. 0 3 Solvig the iequality 0 for d, we obtai the required d o. 5

6 3 Computatioal results Note that the smallest orders of a tree, where Theorem leaves ope the existece of Leech trees, is = 9 ad vertices. Theorem 4. There are o Leech trees o =9ad vertices. We will give the outlie of the algorithm that we use to check the existece of Leech tree o 9 vertices. The oe for is similar. A backtrack algorithm is used to fid whether there exists a Leech tree of 9 odes. The weighted tree of 9 odes will be represeted as a 9 9 adjacecy matrix. There are eight edges for a weighted tree of 9 odes, which ca be labeled as edge through edge 8 such that their weights are i icreasig order. Obviously the weight of edge has to be oe. Ad we ca fix the positio of edge to be the, etry of the matrix. The algorithm will do as follows: Iitial step: set matrix m to be the adjacecy matrix with edge assiged Step i i 8 hadle edge i : weight Fid-Next-Weightm; for all possible positios do: if Check-Validitypositio,m==FALSE, try ext positio; copym i,m i ; Assig-Valuem i,weight,positio; if Is-Validm i ==FALSE, try ext positio; Do step i +; Fid-Next-Weight: fid out the weights of all the existig paths, the take the smallest weight from {,,...,36} that is ot i there, this has to be the weight of the ext edge. 6

7 Check-Validity: sice we are lookig for a tree, we do ot wat cycles, so check the curret graph for all possible paths, the ew edge ca ot be i a positio to coect two ed vertice s of a existig path which will result i a cycle. Is-Valid: check if ay two paths of the existig graph has the same weight. Note that at step 8 whe all the eight edges are assiged, we just eed to check if it represets a valid Leech tree. To fid out the weights of all paths, we eed to calculate the distace matrix of the graph. This was doe usig the classical all-pair shortest path algorithm. If all 8 edges are asisged ad the matrix is still valid, the program eds with a foud Leech tree. The code ca be dowloaded from We ote that Waye Goddard has idepedetly arrived to the same computatioal results. 4 Buema s 4-poit coditio Cosider some vertices of a tree with positive edge weights. Defie the distace of two vertices by the weight of the path coectig the vertices. It is well-kow ad easy to verify that these distaces defie a metric space o the vertex set. This metric space has a peculiar property, the so-called Buema s 4-poit coditio: for ay 4 vertices x, y, u, v, twooutofthe followig three distace sums dx, y+du, v, dx, u+dy, v, dx, v+du, y are equal, ad the third is ot greater tha them. A glimpse o the figure below shows why this holds. I Figure, a, b, c, d, e 0 are the legths of the correspodig paths, ad we have dx, y+du, v =a+b+d+e 7

8 x a b y c u e d v Figure : Buema s 4-poit coditio. a + b +c+d+e=dx, u+dy, v =dx, v+dy, u. Cosider ow the followig problem: which fiite metric spaces ca be represeted i the way described above? Buema s theorem asserts that exactly those fiite metric spaces have such a tree represetatio, for which Buema s 4-poit coditio holds. It is worth otig that the triagle iequality of the metric space also follows from Buema s 4-poit coditio, if it is exteded to ay four ot ecessarily distict poits. Although Buema s 4-poit coditio was actually discovered twice before Buema did, see [5, 7], the result is still little kow amog those who do ot work o phylogeetic tree recostructio. Oe might pose the followig cojecture, stregtheig Cojecture : Cojecture. For ay fiite metric space o > 0 vertices with distaces,, 3,...,, Buema s 4-poit coditio fails. This cojecture would say that,, 3,..., caot be the distace set of vertices i a edge-weighted tree that may have more tha vertices ad ot ecessarily edge weighted with itegers! Note, however, that Taylor s proof to Theorem still works if we allow ew vertices i the tree but require that all ew vertices have iteger distaces to all old vertices. We show a example that Cojecture is false for ay order. For ay, the followig is a example of a tree T whose edge weights are half itegers. There are vertices i V T such that pairwise distace amog these vertices are,, 3,...,. I Figure 3, we assig half-iteger weights to the edges as follows: v,v =, v,v 3 =, v,u =, 8

9 v 3 v 4 v 5 v 6 v 7 v v v v u u u 3 u 4 u 5 u 4 v u 4,v = Figure 3: Example u 4,v ; u i,u i = i for i =,3,..., 4; u i,v i+3 = i+3 + for i =,,..., 4. The, if we cosider the distace matrix for vertex v i,i =,,...,,we have v v v... v v v v v v 0 v 0.. Refereces [] P. Buema, The recovery of trees from measures of dissimilarity, i Mathematics i the Archaeological ad Historical Scieces, Proceedigs of the Aglo-Romaia Coferece, Mamaia, 970, F. R. Hodso, 9

10 D. G. Kedall, P. Tautu, eds.; Ediburgh Uiversity Press, Ediburgh, 97, [] J. Leech, Aother tree labelig problem, Amer. Math. Mothly 8 975, [3] G. Rigel, Problem 5, Theory of graphs ad its applicatios Proc. Sympos., Smoleice, 963 5, Publ. House of Czech. Acad. Sci., Prague, 964, p. 6. [4] H. Salmasia, A result o the prime labeligs of trees, Bull. Ist. Comb. Appl , [5] Y. A. Smolesky: A method for liear recordig of graphs, USSR Comput. Math. Phys., 969, [6] H. Taylor, Odd path sums i a edge-labeled tree, Math. Magazie [7] K. A. Zaretsky, Recostructio of a tree from the distaces betwee its pedat vertices, Uspekhi Math. Nauk Russia Mathematical Surveys, 0 965, 90 9 i Russia. 0