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ЧЕБЫШЕВСКИЙ СБОРНИК
Том 21. Выпуск 2.
УДК 514.7+519.17+519.8 DOI 10.22405/2226-8383-2020-21-2-169-189
Расстояния Громова — Хаусдорфа до симплексов и некоторые приложения к дискретной оптимизации
А. О. Иванов, А. А. Тужилин
Иванов Александр Олегович — профессор, механико-математический факультет, Московский государственный университет имени М. В. Ломоносова; профессор, кафедра ФН-12, Московский государственный технический университет имени Н. Э. Баумана (г. Москва). e-mail: aoivaQmech.math.тsu.su
Тужилин Алексей Августинович — профессор, механико-математический факультет, Московский государственный университет имени М. В. Ломоносова (г. Москва). e-mail: [email protected]
Аннотация
В работе изучается взаимосвязь между расстоянием Громова — Хаусдорфа и задачами дискретной оптимизации. Расстояние Громова — Хаусдорфа до метрического пространства с одинаковыми непутевыми расстояниями используется используется для решения следующих проблем: вычисление длин ребер минимального остовного дерева для конечного метрического пространства; обобщенная пробам Борсука; вычисление хроматического числа и минимального размера клинкового покрытия для простого графа.
Ключевые слова: расстояние Громова — Хаусдорфа, минимальное остовное дерево, проблема Борсука, хроматическое число, клинковое покрытие, метрическая геометрия, дискретная оптимизация
Библиография: 26 названий. Для цитирования:
А. О. Иванов, А. А. Тужилин. Расстояния Громова — Хаусдорфа до симплексов и некоторые приложения к дискретной оптимизации // Чебышевский сборник, 2020, т. 21, вып. 2, с. 169— 189.
CHEBYSHEVSKII SBORNIK Vol. 21. No. 2.
UDC 514.7+519.17+519.8 DOI 10.22405/2226-8383-2020-21-2-169-189
Gromov^Hausdorff Distances to Simplexes and Some Applications to Discrete Optimisation
A. O. Ivanov, A. A. Tuzhilin
Ivanov Alexander Olegovich — professor, Faculty of Mechanics and Mathematics, Lomonosov Moscow State University; professor FN-12, Bauman Moscow State Technical University (Moscow). e-mail: aoivaQmech.math.msu.su
Tuzhilin Alexey Augustinovich — professor, Faculty of Mechanics and Mathematics, Lomonosov Moscow State University (Moscow). e-mail: [email protected]
Abstract
Relations between Gromov-Hausdorff distance and Discrete Optimisation problems are discussed. We use the Gromov-Hausdorff distances to single-distance metric space for solving the following problems: calculation of lengths of minimum spanning tree edges of a finite metric space; generalised Borsuk problem; chromatic number and clique cover number of a simple graph calculation problems.
Keywords: Gromov-Hausdorff distance, Minimum spanning tree, Borsuk problem, chromatic number, clique covering, metric geometry, discrete optimisation
Bibliography: 26 titles. For citation:
A. O. Ivanov, A. A. Tuzhilin, 2020, "Gromov-Hausdorff Distances to Simplexes and Some Applications to Discrete Optimisation" , Chebyshevskii sbornik, vol. 21, no. 2, pp. 169-189.
1. Introduction
The aim of the paper is to demonstrate close connections between the geometry of Gromov-Hausdorff distance and such popular Discrete Optimisation problems as minimum spanning tree problem, Borsuk conjecture, estimation of chromatic number and clique cover number of a simple graph. We start with a short informal review, all necessary formal definitions can be found below.
A general concept of distance is usually used to measure a difference between objects under consideration. Distances have applications in almost all spheres of human activity, from Geography to Linguistics, from Biology to Theology. A great number of beautiful examples can be found in [1]. A natural idea to compare subsets of a given metric space or, more generally, compare different metric spaces using appropriate distances, leads to appearance of so-called hyperspaces, i.e., metric spaces of some spaces, see, for example [2]. For subsets A and B of a fixed metric space X, a natural distance function dn was defined by F. Hausdorff [4] as the infimum of positive numbers r such that A is contained in the r-neighbourhood of B, and vice-versa. It is well-known that this function, referred as the Hausdorff distance, is a metric on the family of all closed bounded subsets of the metric space X, see for example [3]. The Hausdorff distance was generalised to the case of two metric spaces X and Y by D. Edwards [5] and independently by M. Gromov [6]. They suggested to take the infimum of the values dn (<p(X)) over all possible isometrical embeddings ^X ^ Z
and ^Y ^ Z into all possible metric spaces Z. Now this value is referred as the Gromov-Hausdorff distance between X and Y. It is well-known that this distance function a metric on the family of isometrv classes of compact metric spaces. The corresponding hvperspace is usually denoted by M and is referred as the Gromov-Hausdorff space.
The geometry of the Gromov-Hausdorff space is rather tricky and is intensively investigated by many authors, see a review in [3]. Recently the technique of closed optimal correspondences permitted to prove that the space M is geodesic [7], to describe some local and all global isometries of M, see [8] and [9]. Since finite metric spaces form an everywhere dense subset of M, the distances to such spaces and between such spaces play an important role in the research of geometry of M. Important classes of such spaces are formed by the ones all whose non-zero distances are the same (so-called single-distance spaces or simplexes) and by the spaces whose non-zero distances take only two different values (so-called two-distance spaces). The authors, together with S. Illiadis and D. Grigor'ev, see [10], [11], calculated distances from any metric space to any simplex, and, as a particular case, the distances between any simplex and any 2-distance space, see [12]. It turns out that the Gromov-Hausdorff distance from a metric space X to a simplex "feels" somehow a geometry of partitions of the space X. The latter explains some relations between the Gromov-Hausdorff distance and Discrete Optimisation problems.
Many Discrete Optimisation problems are related to Geometry, have a long history, and are either unsolved yet, or solved only in some particular cases. Due to many natural applications and total computerisation Discrete Optimisation is one of the most fast developing branch of modern Mathematics. Describe shortly the problems considered in the paper. Start with a problem of metric minimum spanning trees.
For a finite subset M of a metric space X, consider the complete graph K(M) with the vertex set M, endowed with the weight function whose value on an edge [x, y} equals to the distance Ixyl between the points x and y in the space X. A minimum spanning tree on M is a subtree of K(M) with the same vertex set M and the least possible total weight. It is well-known that such a tree can be always constructed (even in a polynomial time) by a greedy algorithm such as the Kruskal algorithm [13]. Generally speaking, a minimum spanning tree on a fixed subset MCl is not defined uniquely, but the ordered list of the weights of edges is the same for all such trees. This list is referred as an mst-spectrum of M. It is shown, see Section 4.1 and paper [25], that the mst-spectrum of M can be calculated in terms of the Gromov-Hausdorff distance from M to the simplexes consisting of k = 2,..., #M points and such that there diameters are sufficiently large (they has to be at least twice greater than the diameter of M).
Now, let us pass to Borsuk Problem. In 1933, a Polish mathematician Karol Borsuk asked the following question: How many parts one needs to partition an arbitrary subset of the Euclidean space into, to obtain pieces of smaller diameters? He made the following famous conjecture: Any bounded non-single-point subset of Rn can be partitioned into at most n + 1 subsets, each of which has smaller diameter than the initial subset. K. Borsuk himself proved it for n = 2 and for a ball in 3-dimensional space, [14] and [15]. Next, the conjecture was proved by J. Perkal (1947), and independently, by H.G. Eggleston (1955) for n = 3, then in 1946 by H. Hadwiger [18] and [19] for convex subsets with smooth boundaries, then for central symmetric bodies by A. S. Riesling (1971), and to this moment almost everybody believed that it is true. However, in 1993 the conjecture was suddenly disproved in general case by J. Kahn, and G. Kalai, see [20]. They constructed a counterexample in dimension n = 1325, and also proved that the conjecture is not valid for all n > 2014. This estimate was consistently improved by Raigorodskii, n ^ 561, Hinrichs and Richter, n ^ 298, Bondarenko, n ^ 65, and Jenrich, n ^ 64, see details in a review [21]. Notice that all the examples are finite subsets of the corresponding spaces, and the best known results of
2
On the other hand, Lusternik and Schnirelmann [16], and a bit later independently Borsuk [14]
and [15], see also [17], have shown that the standard sphere and the standard ball in Rra, n ^ 2, cannot be partitioned into m ^ n subsets having smaller diameters. Thus, the least possible number of parts of smaller diameter, necessary to partition the sphere and the ball in Rra equals n + 1.
In the present paper we consider a generalized Borsuk problem, passing to an arbitrary bounded metric space X and its partitions of an arbitrary cardinality m (not necessary finite). We give a criterion solving the Borsuk problem in terms of the Gromov-Hausdorff distance. It is shown that to verify the existence of an m-partition into subsets of smaller diameter it suffices to calculate the Gromov-Hausdorff distance from the space X to a simplex having the cardinality m and a smaller diameter than X, see Section 4.2. As a corollary, a solution to the Borsuk problem for a 2-distance space X with distances a < b is obtained in terms of the clique cover number of the simple graph G with vertex set X, whose vertices x and y are connected by an edge iff Ixyl = a.
Recall that a clique cover of a given simple graph is a cover of the vertex set of the graph by subsets within which every two vertices are adjacent. Each such subset is called a clique and is a vertex set of a complete subgraph that is also referred as a clique. The minimum k for which a fc-element clique cover exists is called the clique cover number of the given graph. Further, a graph coloring is an assignment of labels traditionally called "colors" to vertices of a graph in such a way that no two adjacent vertices are of the same color. The smallest number of colors needed to color a graph is called its chromatic number. It is well-known that the clique cover can be considered as a graph coloring of the dual graph, hence the clique cover number of a graph equals to the chromatic number of the dual one. Calculation and estimation of these numbers are very hard combinatorial problems related to many other problems of Discrete Optimisation, in particular, to Borsuk conjecture, see a review in [24]. We calculate the clique cover number of a simple graph and the chromatic number of a simple graph in terms of the Gromov-Hausdorff distance from an
2
The work is partly supported by RFBR, Project 19-01-00775-a, and by MGU Scientific Schools Support program.
To conclude this short Introduction, the authors use the opportunity to congratulate our Teacher, Anatolv Timofeevich Fomenko, on his 75th birthday and wish him good health, beautiful results and many birthdays ahead. He stimulated us to become professional mathematicians, and teaches us to work hard, to live in the World of Mathematics, and to be optimistic both in science and in life. We are infinitely thankful for his deep influence, kind care, permanent support and attention.
2. Preliminaries
Let X be an arbitrary nonempty set. Recall that a function on pXxX ^ R is called a metric if it is non-negative, non-degenerate, symmetric, and satisfies the triangle inequality. A set with a metric is called a metric space. If such a function p is permitted to take infinite values, then we call p a generalized metric. If we omit the non-degeneracv condition, i.e., permit p(x,y) = 0 for some distinct x and y, then we change the term "metric" to pseudometric. If p is only non-negative, symmetric, and p(x, x) = 0 for any x £ X, then we call such p a distance function, instead of metric or pseudometric. As a rule, if it is not ambiguous, we write for p(x,y).
In what follows all metric spaces are endowed with the corresponding metric topology. We also use the following notations. By #X we denote the cardinality of a set X. Let X be a metric space. The closure of a subset AcX is denoted by A For its arbitrary nonempty subset AcX and point x £ X put \xA\ = IAx\ = inf {\ax\ : a £ A}. Further, for r ^ 0 put
Br(x) = {y £ X : \xy\ ^ r], and Ur(x) = {y £ X : \xy\ < r],
and
Вг(А) = {у е X : lAyl < г}, and (А) = {у е X : |Ау| < г}.
2.1. Hausdorff distance
Recall the basic results concerning the Hausdorff distance. The details can be found in [3]. For a set X, by V0(X) we denote the collection of all nonempty subsets of X. Let X be a metric space. For any A, B £ V0(X) we put
It is well-known that these three values coincide with each other, i.e., dlH(A,B) = d2H(AB) = = d3H(A, B) for any A,B £ V0(X). The value dlH(A, B) is denoted by dn(A, B). It is easy to see that dn is non-negative, svmmetric, and dn(A, A) = 0 for any nonempty AcX, thus, dn is a generalized distance on the family Vq(X) of all nonempty subsets of a me trie space X, moreover, it is a generalized pseudometric on V0(X), i.e., it satisfies the triangle inequality. The function dn is referred as Hausdorff distance.
Further, by %(X)cV0(X) we denote the set of all nonempty closed bounded subsets of a metric space X. ft is well-known that the Hausdorff distance dn is a metric on %(X).
In what follows, speaking about the distance in %(X) we will always mean the Hausdorff distance. Notice that there are different notations for this hvperspace in the literature. We use the notation %(X) by virtue of the fact that this is the largest natural set of subsets of a metric space which the Hausdorff distance is a metric on.
Recall a few properties of the Hausdorff distance.
Proposition 1. Let, X he an arbitrary metric space.
1. The mapping f X —y ~Pq(X) given by the formula f x — [x} is an isometric embedding.
2. For any A,B £ V0(X) we have dH(A, B) = dH(A, B) = dH(A, B) = dH(A, B).
3. For any A,B £ Vq (X) we ha ve dH (A, B) = 0 if and on! y if A = B.
4- IfYCX is an e-net in AcX, then dn(A, Y) ^ e.
2.2. Gromov^Hausdorff distance
Let X and Y be metric spaces. A triple (X', Y', Z) consisting of a metric space Z and its two subsets X 'and Y' which are isometric respectively to X and Y is be called a realization of the pair (X,Y). Put
dGH(X, Y) = inf {r £ R : 3 a realization (X', Y', Z) of (X, Y) such that dH(X', Y') < r}.
Remark 1. The value den(X,Y) is evidently non-negative, symmetric, and den(X,X) = 0 for any metric space X. Thus, dan is a generalized distance function on each set of metric spaces.
Definition 1. The value dan(X,Y) is called the Gromov-Hausdorff distance between the metric spaces X and Y.
dlH (A,B) = max (sup {|аБ| : а е A), sup{ |A6| : b е B}^J,
d2H (A, B) = inf {г е [0, те] : AcBr (В) & Br (A)B }, d3H (A, B) = inf {г е [0, те] : AcUr (В) & Ur (A)B }.
(1) (2) (3)
It turns out that, to define the Gromov-Hausdorff distance, it suffices to consider only metric spaces of the form (X U Y, p), where p extends the original metrics of ^^d Y, i.e., the restrictions of p onto X and Y coincide with the original metrics of these metric spaces. Such p is called an admissible metric for X and Y, and the set of all admissible metrics for given X and Y is denoted bv V(X, Y).
Proposition 2. For any metric spaces X and Y, we have
dGH(X,Y) =inf{pH(X,Y) : p £V(X,Y)}. (4)
It is well-known that, on every set of metric spaces, the function dan is a generalized pseudometric. If the diameters of all spaces in the family are bounded by the same number, then don is a pseudometric. In general, dan is not a metric, it may equal zero for distinct metric spaces. However, if we restrict ourselves to compact metric spaces considered up to an isometrv, then dan is a metric.
For specific calculations of the Gromov-Hausdorff distance, other equivalent definitions of this distance are useful.
Recall that a relation between sets X and Y is defined as a subset of the Cartesian product XxY. Similarly to the case of mappings, for each a £ V0(XxY) and for every x £ X and y £ Y, there are defined the image a(x) := {y £ Y : (x,y) £ ct} of any x £ X and the pre-image a-1(y) = {x £ X : (x,y) £ ct} of any y £ Y. Also, for AcX and BCY their image and pre-image are defined as the union of the images and pre-images of their elements, respectively.
Let kx XxY ^ X and ■ky XxY ^ Y be the canonical projections, i.e., ^x(x,y) = x and ■ky(x,y) = y. The restrictions of these mappings to each relation aCXxY are denoted in the same way. A relation R between X and Y is called a correspondence if the restrictions of the canonical projections ^x and ■ky onto R are surjective. In other words, for every x £ X there exists y £ Y, and for every y £ Y there exists x £ X, such that (x, y) £ R. Thus, the correspondence can be considered as a surjective multivalued mapping. By ~R(X, Y) we denote Tthe set of all correspondences between X and Y.
If X and Y are metric spaces, then for each relation a £V0(XxY) its distortion disa as follows
disCT = sup{ |\xx'\ - \yy'11 : (x, y), (x', y') £ a^
Remark 2. For any a1, a2 £ V0(X xY) such th at a1Ca2, we ha ve disoi ^ disa2.
The next constructions establish a link between correspondences from ~R(X, Y) and admissible metrics on X U Y. At first, let p £ V(X,Y) be an arbitrary admissible metric for metric spaces X and Y, and suppose that pn(X,Y) < <x. Choose an arbitrary r ^ pn (X,Y) such that the set Rr = {(x, y) : p(x, y) ^ r} is a correspondence between X and Y (it is so for any r > pn(X, Y)). Then disR? < 2r.
Conversely, consider an arbitrary correspondence R £ ~R(X, Y). Suppose that disR < x>. Extend the metrics of X and Y up to a symmetric function pR defined on X U Y as follows:
pR(x,y) = pR(y,x) = inf{\xx'\ + \yy'\ + ^disE : (x',y') £ R}.
If disR > 0 then pR is an admissible metric, and p^ (X, Y) = 2disR.
The key well-known result on the relation between the correspondences and the Gromov-Hausdorff distance is the following Theorem.
Theorem 1. For any metric spaces X and Y the equality
dGH (X, Y) = 1 inf {disR : R £ K(X, Y)}
holds.
2.3. Irreducible correspondences
For arbitrary nonempty sets X and Y, a correspondence R £ /(X, Y) is called irreducible if it is a minimal element of the set /(X,Y) with respect to the order given by the inclusion relation. The set of all irreducible correspondences between X and Y is denoted by /°(X, Y). The following result is evident.
Proposition 3. A correspondence R £ /(X, Y) is irreducible if and only if for any (x, y) £ R it holds
Res{#R(x), #R-1(y)} = 1.
Theorem 2. Let X, Y be arbitrary nonempty sets. Then for every R £ /(X, Y) there exists R° £ n°(X, Y) such that R°CR. In particular, K°(X, Y) = 0.
Theorems 2 and 1, together with Remark 2, implies
Corollary 1. For any metric spaces X and Y we have
dGH(X,Y) = 1 inf{disR | R £ K0(X,Y)}.
Now we give another useful description of irreducible correspondences.
Proposition 4. For any nonempty sets X, Y, and each R £ /°(X,Y), there exist and unique partitions Rx = [X^^i and Ry = [Yi}i^i of the sets X and Y, respectively, such that R = UiElXixYi. Moreover, Rx = Uyey{R-1 (y)}, Ry := Ux€x{R(x)},
[XixYi}iEl = U{x,y)^R[R-1(y)xR(x)},
and for each i it hoids Res [#Xi, #Yi} = 1.
Conversely, each set R = UieiXixYi, where [Xi}ie^ rnd [Yi}iei are partitions of nonempty sets X and Y, respectively, such that for each i it holds Res[#Xi, #Yi} = 1; is an irreducible correspondence between X and Y.
Let X be an arbitrary set consisting of more than one point, and m a cardinal number, 2 ^ m ^ By Vm(X) we denote the family of all possible partitions of the set X into m nonempty subsets.
Now let X be a metric space. Then for each D = [Xi}iej £ Vm(X) we put
diam^ = supdiamXj. iei
Further, for any nonempty A,BcX, we put IABI = inf{labl : (a,b) £ AxB}, rnd \AB\' := sup{labl : (a,b) £ AxB}. Further, for each D = [Xi}iej £ Vm(X) we put
a(D) = inf{lXiX31 : i = j} and $(D) = sup^X,|' : i = j}.
Also notice that |XiXi| = 0 lXiXj\' = diamXj, and hence, diam^ = supie/ lXiXi|'.
The next result follows easily from the definition of distortion, as well as from Proposition 4.
Proposition 5. Let, X and Y be arbitrary metric spaces, Dx = [Xj}^, Dy = [Yi}iej, #1 ^ 2, be some partitions of the spaces X and Y, respectively, and R = UieiXixYi £ /(X,Y). Then
disE = sup{|XiX,|' - lYY)l, \YiYj|' - lXiX31 : i,j £ I} =
= sup{diam^x, diam^Y, \XiXj- \YiYjlYiYj- \XiXj\ : i,j £ I, i = j] ^
^ max{diam^x, diam^Y,@(Dx) - a(DY(DY) - a(Dx)}.
It will also be convenient for us to represent a relation a £V0(XxY) as a bipartite graph. Then the degree deg of each vertex is defined: degCT(x) = #a(x) and degCT(y) = #a-1(y).
Remark 3. Notice that if R £ R°(X,Y), x £ X, and degR(x) > 1, then for each x' £ X, x' = x, it holds R(x) n R(x') = 0. Therefore, if #X ^ 2 and #Y ^ 2, then for any R £ R°(X, Y) there is no x £ X such that {x}xYCR.
2.4. Some Examples and Estimates
Here we list several simple cases of exact calculation and estimate of the Gromov-Hausdorff distance.
Example 1. Let Y be an arbitrary e-net of a metric space X. Then den(X, Y) ^ dn(X, Y) ^ e. Thus, every compact metric space is approximated (according to the Gromov-Hausdorff metric) with any accuracy by finite metric spaces.
By A1 we denote a single-point metric space.
Example 2. Then for any metric space X we have
don (A1,X) = -diamX.
Example 3. Let X and Y be some metric spaces, and the diameter of one of them is finite. Then
dGH (X, Y) ^ - |diamX - diamF Example 4. Let X and Y be some metric spaces, then
don (X,Y) ^ - max{diamX, diamF},
in particular, if X and Y are bounded metric spaces, then dan (X,Y) <
For an arbitrary metric space X and a real A > 0, by XX we denote the metric space obtained from X by multiplying all distances by A. For A = 0 we set XX = A1.
Example 5. For any bounded metric space X and any X ^ 0 № ^ 0 we have den(XX, pX) = = 1 \X — diamX, in particularly, for any 0 ^ a < b the curve 7(t) := tX, t £ [a, b], is shortest.
Example 6. Let, X and Y be metric spaces, then for any X > 0 we ha ve den (XX, XY) = = XdcH(X,Y). If', in addition, dan(X,Y) < m, then the equality holds for all X ^ 0.
3. Gromov-Hausdorff Distance to Simplexes
By simplex we call a metric space, all whose non-zero distances equal to each other. If m is an arbitrary cardinal number, then by ATO we denote a simplex containing m points and such that all its non-zero distances equal — Thus, XAm, X> 0, is a simplex whose non-zero distances equal A. Also, for arbitrary metric space X and A = 0, the space XX coincides with A1 by definitoon.
3.1. The Case of Simplexes of Greater Cardinality
The next result generalizes Theorem 4.1 from [10].
Theorem 3. Let X be an arbitrary metric space, m > #X a cardinal number, and X ^ 0; then
2dcH (XAm,X) = max{A, diamX — X}.
proof. If X is unbounded, then 2dcn(XAm,X) = to bv Example 3, and the required equality holds.
Now, let diamX < to.
If #X = 1, then diamX = 0, and, by Example 2, we have
2dcH (XA,X) = diamAA = A = max{A, diamX — X}.
If A = 0, then, by Example 2, we have
2dcH (Ai ,X) = diamX = max{A, diamX — X}.
Let #X > 1 and A > 0. Choose an arbitrarv R £ ~R(XAm,X). Since #X < m and A > 0, then there exists x £ X such that #R-1(x) ^ 2, Aus, disE ^ ^d 2dcn(XAm,X) ^ A.
Consider an arbitrary sequence (xi,yi) £ X xX such th at Ixiyil ^ diamX. If it contains a subsequence (xik,yik) such that for each ik there exists Zk £ XA, (zk,Xik) £ E, (zk,yik) £ E, then disE ^ diamX and
2dcH(XAm,X) ^ max{A, diamX} ^ max{A, diamX — X}.
If such subsequence does not exist, then there exists a subsequence (xik ,yik) such that for any ik there exist distinct Zk,Wk £ XAm, (zk,Xik) £ E, (wk,yik) £ E, and, therefore,
2dcH(XAm,X) ^ max{A, |diamX — A^ ^ max{A, diamX — X}. Thus, in the both cases we have 2dcn(XA,X) ^ max{A,diamX — A}.
Choose an arbitrary x0 £ X, then, by assumption, #X > 1, and, thus, the set X\{^o} is not empty. Since #X < m, then XAm contains a subset XA' of the same cardinality as X\{^0}. Let g XA' ^ X\{^0} be an arbitrary bijection, and XA'' = XAm\XA', then #AA'' > 1. Consider the following correspondence
Eo = {(z',g(z')) : z' £ AA'} U (AA''x{^o}) and apply Proposition 5. So, we have:
disE0 = sup{A, lx1x'1\ — X,X — lx2 x'2l : x1, x1, x2, x'2 £ X, x1 = x'i, x2 = x'2} = max{A, diamX — A}, therefore,
2dGH(XA, X) = max{A, diamX — X},
what is required. □
3.2. The Case of Simplexes with at most the Same Cardinality
Let X be an arbitrary set consisting of more th an one point, 2 ^ m ^ #X a cardinal number, and A > 0. Under notations of Subsection 2.3, consider an arbitrary D £ Vm(X), any bijection g XAm — D, and construct the correspondence Rd £ /(XAm, X) in the following way:
Rd = |J [z}xg(z).
ze\Am
Clearly that the correspondence Rd is irreducible. Apply Proposition 5 to calculate its distortion.
Proposition 6. Let, X = A1 he an arbitrary metric space, 2 ^ m ^ #X a cardinal number, and X > 0. Then for any D £ Vm(X) it holds
disRo = max[diam^, A - a(D), ft(D) - X}.
Proof. If X is unbounded, then disE = to for any R £ /(XAm,X). Since m ^ 2, for any D = [Xi}iej £ Vm(X) we have either diam^ = to, or fi(D) = to. Indeed, if diam^ < to and P(D) < to then for any x,y £ X either x,y £ X^ thus \xy\ ^ diam^, or x £ X^ y £ Xj, i = j, and \xy\ ^ \XiXj^ ft(D), therefore X is bounded. Thus, for an unbounded X the both sides of the equality are infinite, thus we get what is required. Now, let diamX < to. By Proposition 5, we have
disE^ = supjdiam^, A -\XiXj \XiXj- X: i,j £ I,i = j} = max[diam^, A - a(D), ft(D) - A},
that completes the proof. □
Corollary 2. Let X = A1 be an arbitrary metric space, 2 ^ m ^ #X a cardinal number, and X > 0. Then for any D £ Vm(X) it holds
disRo = max[diam^, A - a(D), diamX - X}.
PROOF. For unbounded X the equation evidently holds.
Consider now the case of bounded X. Notice th at diam^ ^ diamX and ft (D) ^ diamX. In addition, if diam^ < diamX, and (xi,yi) £ XxX is a sequence such that lxiyi\ — diamX, then, starting from some i, the points s^d yi belong to different elements of D, therefore, in this case we have fi(D) = diamX, and the formula is proved.
Now, let diam^ = diamX, then fi(D) - X ^ diamX and diamX - A ^ diamX, thus
max[diam^, A - a(D), fi(D) - X} = max[diamX, A - a(D)} = max[diam^, A - a(D), diamX - A},
□
Proposition 7. Let X = A1 be an arbitrary metric space, and 2 ^ m ^ #X a cardinal number, and X > 0. Then
2dGH (XAm,X )= inf disED.
Devm(x)
proof. The case of unbounded X is trivial, so, let X be bounded. By Corollary 1,
2dGH (XAm,X )= inf disE,
Ren0(\Am,x)
thus, it suffices to prove that for any irreducible correspondence R £ ~R°(XAm,X) there exists D £ Vm(X) such that disEo ^ disE.
Let us choose an arbitrary R £ R0(\Am,X) such that it cannot be represented in the form Rd, then the part ition is not pointwise, i.e., the re exists x £ X such that #R-1(x) ^ 2,
therefore, disE ^ A.
Define a metric on the set D^ to be equal A between any its distinct elements, then this metric space is isometric to a simplex XA'n, n ^ m. The correspondence R generates naturally a correspondence E' £ R(\A'n,X), namely, if D^ = {Aj}jej, and fuDRAm ^ is the bijection generated by E, then
R' = |J {Aj }xfR(A3). jeJ
It is easy to see that disE = max{A, disE'}. Moreover, R' is generated by the partition D' = i.e., R' = RD', thus, by Corollary 2, we have
disE' = max{diam^', A — a(D'), diamX — A},
and hence,
disE = max{A, diam^', A — a(D'), diamX — X} = max{A, diam^', diamX — X}. Since n ^ m, the partition D' has a subpartition D £ Vm(X). Clearly, diam^ ^ diam^', therefore,
disED = max{diam^, A — a(D), diamX — X} ^ max{diam^', A, diamX — X} = disE, q.e.d. □
Considering separately the trivial case of A = 0, we get the following result.
Corollary 3. Let X = A1 be an arbitrary metric space, 2 ^ m ^ #X a cardinal number, and X ^ 0. Then
2dah(XAm, X) = inf max{diam^, A — a(D), diamX — X}.
Devm(x)
For any metric space X put
e(X) = inf{\xy\ : x,y £ X, x = y}.
Notice that e(X) ^ diamX, and for a bounded X the equality holds if and only if X is a simplex. Corollary 3 immediately implies the following result that is proved in [10].
Theorem 4 ([10]). Let X = A1 be a finite metric space, m = #X, and X ^ 0, then
2dG H (XAm, X) = max {A — e(X), diamX — X}.
4. Some Applications
In this section we apply the previous results to some well-known discrete optimisation problems from Metric Geometry and Graph Theory.
4.1. Calculation mst-spectrum
The first application deals with optimal graphs, so we start from some preliminaries for the Graph Theory.
4.1.1. Elements of Graph Theory
Here we consider simple graphs only, so in what follows bv a graph we mean a pair G = (V,E) consisting of two sets V and E referred as the vertex set and the edge set of the graph G , respectively; elements of V are called vertices, and the ones of E are called edges of the graph G. The set E is a subset of the family of two-element subsets of V. If V and E are finite sets then the graph G is called finite.
It is convenient to use the following notations:
• If [v, w} £ E is an edge of the graph G, then we write it just as vw or ww, further one says that an edge vw connects the vertices v and w, and that v and w are the vertices of the edge vw,
• We write V(G) and E(G) for the vertex set and the edge set of a graph G to underline which graph is under consideration.
Graphs G = (V, E^d H = (W, F) are called isomorphic if there exists a bijective map fV — W such that uv £ E if and only if f (u)f (v) £ F. Such a mapping f is called an isomorphism of the graphs G and H. Isomorphic graphs are often identified and, therefore, are not distinguished.
Two vertices v,w £ V(G) are called adjacent if vw £ E(G). Two different edges e1,^ £ E(G) are called adjacent if they have a common vertex, i.e., if e1 fl ^ = 0. Each edge vw £ E(V) and its vertex, i.e., v or w, are said to be incident to each other. The set of vertices of a graph G adjacent to a vertex v £ V is called the neighbourhood of the vertex v and denoted bv Nv. The cardinal number of edges incident to a vertex v is called the degree of the vertex v and is denoted bv deg v, so deg v = #Nv.
A subgraph of a graph G = (V, E) is each graph H = (W, F) provided that WCV md FCE. The fact that a graph H is a subgraph of a graph G is denoted as HCG. li W = V then the subgraph HCG is called spanning.
On the set of all graphs, whose vertex sets lie in a given set V, the inclusion relation c of being
(0, 0)
greatest one is called the complete graph on V and is denoted by K(V). This partial order induces the one on the set of all subgraphs of a graph G = (V,E): now the smallest element is again the empty graph (0,0), but the greatest one is the graph G itself.
We also need some set-theoretical operations on graphs. They are usually defined in an intuitively clear way in terms of vertex and edge sets. For example, if G1 = (V1,E1) and G2 = (^2,^2) are graphs, then put G1U G2 = (V1U V2, E1U E2)• Also, if G = (V, E) is a gra^ and e is a two-element subset of F, then G U e = (V,E U {e}); similarly for e £ E put G\e = (V., E\[e}).
For each WCV the subgraph G(W) of the graph G generated by W is defined as the graph with the vertex set W, whose edge set consists of all e £ E that connects vertices from W. In other words, G(W) is maximal among subgraphs of G, whose vertex sets coincides with W.
We also need a similar construction for an edges set. Namely, for FCE the subgraph G(F) of the graph G generated by F is defined as the graph with the edge set F, whose vertex set is the collection of all vertices of G incident to edges from F.
A finite sequence 7 = (v0 = v,v1,... ,Vk = w) of vertices of a graph G is called a walk of length k connecting v and w if for every i = 1,..., ^ ^^^ wrtices vi-^ md v,i are adjacent, and the edges e,i = vi-1vi ^re called the edges of the walk 7. A walk containing at least one edge is called non-degenerate, and the walk containing no edges, i.e., with k = 0, is called degenerate. The walk is called closed if v0 = vn, and it is called open otherwise. A trail is a walk with no repeated edges, a path is an open trail with no repeated vertices. A circuit is a closed trail, and a cycle is a circuit with no repeated vertices.
A graph G is called connected if each pair of its vertices are connected bv a walk. Maximal (by inclusion) connected subgraphs of a graph G are called components of G. A graph without cycles is called a forest, and a connected forest is called a tree.
A weighted graph is a graph G = (V, E) equipped with a weight function w E ^ [0, m) (sometimes it is useful to consider more general weight functions, for instance, the ones taking negative or/and negative values also). A weighted graph is denoted by (V,E,u) or (G,w). The weight us(H) of a subgraph HCG is the sum of the weights of all the edges from this subgraph: u(H) = ^eeE(H) w(e). This definition can be extended to trails, in particular, to paths, circuits and cycles, considered as the corresponding subgraphs of G. In the case of a walk 7 = (v0 = v,v1,... ,Vk = w), its weight is defined as the sum of weights of all its consecutive edges: w(l) = Y^n=1 £ (vi-1yi)- F°r graphs without weight functions these notions are defined as well by 1
Remark 4. As in the case of metric spaces, we sometimes won't denote the weight function explicitly. Instead of that, speaking about a weighted graph G, we denote the weight of an object x just by \x\. For example, for e £ E by \e\ we mean the weight of this edge, and for a subgraph HCG by \HI we denote the weight of H, etc.
4.1.2. Minimum Spanning Tree Problem
Let M be a metric space. We consider M as a weighted complete graph K(M) whose weight function equals to the distance between the corresponding edges. By T(M) we denote the set of all spanning trees in K(M). Put
mst(M) = inf \T I
t eT(M)
and call this value by the length of minimum, spanning tree on M. Each T £ T(M) with \T| = mst(M) is call a minimum spanning tree on M. The set of all minimum spanning trees on M is denoted by MST(M).
Remark 5. If M is finite, them MST(M) = 0. For infinite M the situation is rather more difficult, see [26].
Example 7. If all nonzero distances in M are the same, then every spanning tree in K(M) is minimum,, so MST(M) = T(M).
If #M = 3; then each m,inim,u,m, spanning tree is obtained from, the complete graph K(M) by deleting the longest, edge (any of them if there are several).
mst
In this Section we consider only finite metric spaces M, i.e., #M < m.
Notice that a minimum spanning tree, generally speaking, is not uniquely defined. For G £ MST(M), by a(G) we denote the vector whose elements are the lengths of the edges of the tree G sorted in descending order. The following result is well-known, however, we present its proof for completeness.
Proposition 8. For any G1,G2 £ MST(M) the equality a(G{) = a(G2) holds.
PROOF. Recall the standard algorithm for converting one minimum spanning tree to another [13].
Let G1 = G2, Gi = (M, Ei), then E1 = E2 and #E1 = #E2, therefore, there exists e £ E2\E1. The graph G1 U e has a cycle C containing the edge e, and the cycle C does not contain an edge longer than e, because G1 £ MST(M) otherwise. The forest G2\e consists of two trees whose vertex sets we denote by V' and V". Clearly, M = V' U V". The cycle C contains an edge e' = e connecting
a vertex from V' with a vertex from V''. This edge does not lie in E2, otherwise G2 would contain a cycle. Therefore, e1 £ E1\E2.
The graph G2 U e' also contains some cycle C'. By the choice of e', the cycle C' also has the edge e. Similarly to the above, the length of the edge e is less than or equal to the length of the edge e', otherwise G2 £ MST(M). Therefore, \e\ = \e'|.
Replacing the edge e' in G1 with e, we get a tree G'1 of the same length, i.e., it is a minimum spanning tree as well, and G[ and G2 have one common edge more than the trees G1 and G2. Thus, in a finite number of steps, we rebuild the tree G1 into the tree G2, passing through minimum spanning trees. It remains to notice that a(G'{) = a(G1), therefore, a(G1) = a(G2). □ Proposition 8 motivates the following definition.
Definition 2. For any finite metric space M, by a(M) we denote a(G) for an arbitrary G £ MST(M) and call it the ms^^^^^^ta of the space M.
Theorem 5. Let, M be a finite metric space and a(M) = (a1,..., an-1). Then
ak = max{«(^) : D £ Vk+1(M)}.
Proof. Let G = (M,E) £ MST(M) and the set £ be ordered so that |ei| = ai. Denote by D = {M1,..., Mk+1} the partition of the set M into the sets of vertices of the trees forming the forest G\{ei}k=1.
Lemma 1. We have a(D) = |ek|.
PROOF. Indeed, choose arbitrary Mi and Mj, i = j, take arbitrary points ^ and aj in them, respectively, and let 7 be the unique path in G, connecting ^ and a,j. Then 7 contains some edge ep, 1 ^ p ^ k. However, due to the minimality of the tree G, we have
Wiaj | ^ ^ ^ Res ^i | = ^k l
thus Mj| ^ ^|, so a(D) ^ ^|. On the other hand, the edge ek connects some Mp and Mq, then we get a(D) ^ |MpMq | = ^kc |. □
Now consider an arbitrary partition D' = {M1,..., M'k+1}.
Lemma 2. We have a(D') ^ a(D).
PROOF. Due to Lemma 1, it suffices to show that a(D') ^ ^k|. Denote by E' the set consisting of all edges ep £ E, each of which connects some M[ and Mj, i = j. Since G is connected, then the set E' consists of at least A; edges. On the other hand, if some M[ and M'-, i = j, are connected by
an edge e' £ E', then ^[M^| ^ ^|, hence a(D') = Res ^[M^| ^ Rese'eE' ^J| ^ ^\. □
□
mst
In the present section we show that the mst-spectrum of an arbitrary n-point metric space X can be represented as a linear function on the Gromov-Hausdorff distances from this space to the \A2,...,\An for A ^ 2diamX.
Theorem 6. Let X be a finite metric space, a(X) = (a1,..., an-1), A ^ 2diamX. Then
ak = A — 2 dGH (XAk+1,X).
Proof. Choose any 1 ^ k ^ n - 1 and arbitrary irreducible correspondence R £ R°(XAk+1,X). By Proposition 4, there exists partitions R\Ak+1 = [Zi}1^=1 and Rx = [Xi}1^=1 of A A k+1 and X, respectively, such that R = UT^=1ZixXi, and Res[#Zi, #Xi} = 1 to all i. By Proposition 5, it holds disE ^ max[diamEaafc+1, diamEx}. Thus, if for some i we have #Zi > 1, then disE ^ A ^ 2diamX. Since k + 1 ^ n, there exists R such that #Zi = 1 to all i. For such
E, again by Proposition 5, we have disE ^ diamX. Therefore, inf^n^aafc+^x) disE is achieved
/
Now, if R £ R, then it consists of p = k + 1 elements, and Rx £ Vk+1(X). By Proposition 5, we have
disE = sup{diamEx, \XiXj- \, A - \XiXj \ : 1 ^ i<j ^ k + 1} =
= sup {A - \XiXj \ : 1 < i<j < k + 1} = A - a(Rx),
where the second equality holds because for A chosen the estimate
max{\XiXj- X, diamEx} ^ diamX ^ A - diamX ^ A - \XiXj\
holds for any 1 ^ i < j ^ k + 1. Corollary 1, together with above considerations, gives us
2dah(XAk+1,X) = Res disE = Res (X - a(Rx)) = A - max a(D), + 7 Ren Ren " Devk+1(x)
where the last equality holds because each .D generates some R £ ~R.lt remains to apply Theorem 5 saving that max{a(D) : D £ Vk+1(X)} = ak, thus, 2dah(XAk+1,X) = A - ak. □
Corollary 4. Let X be a finite metric space and X ^ 2diamX; then
#x-1
msiX = A(#X - 1) - 2 ^ dGH(AA k+1 ,X).
k=1
4.2. Generalized Borsuk Problem
Classical Borsuk Problem deals with partitions of subsets of Euclidean space into parts having smaller diameters. We generalize the Borsuk problem to arbitrary bounded metric spaces and partitions of arbitrary cardinality. Let X be a bounded metric space, m a cardinal number such that 2 ^ m ^ #X, and D = [X^iej £ Vm(X). We say that D is a partition of X into subsets having strictly smaller diameters, if there exists e > 0 such that diamXi ^ diamX - e fa all i £ I.
The Generalized Borsuk Problem: Is it possible to partition a bounded metric space X into a given, probably infinite, number of subsets, each of which has a strictly smaller diameter than XI We give a solution to this Problem in terms of the Gromov-Hausdorff distance.
Theorem 7. Let, X be an arbitrary bounded metric space and m a cardinal number such that 2 ^ m ^ #X. Choose an arbitrary number 0 < A < diamX, then X can be partitioned into m subsets having strictly smaller diameters if and only if 2dch(XAm,X) < diamX. If not, then 2dGH (XAm,X) = diamX.
PROOF. Due Corollary 3, for the A chosen the inequalitv 2dah(XAm,X) ^ diamX is valid, and the equality holds if and only if for each D £ Vm(X) we have diam^ = diamX. The latter means that there is no partition of the space X into m parts having strictly smaller diameters. □
Corollary 5. Let, d > 0 be a real number, and m ^ n cardinal numbers. By Mn we denote the set of isometry classes of bounded metric spaces of cardinality at most n, endowed with the Gromov-Hausdorff distance. Choose an arbitrary 0 < X < d. Then the intersection
Sd/2(A{) П Sd/2(XAm)
of the spheres (here the the spheres are considered as spheres in Mn) does not contain spaces, whose cardinality is less than m, and consists exactly of all metric spaces from M.n, whose diameters are equal to d and that cannot be partitioned into m subsets of strictly smaller diameters.
Proof. Let X belong to the intersection of the spheres, then diamX = d in accordance with Example 2. If m > #X, then, due to Theorem 3, we have
2 doh( XAm,X) = max{A, diamX — X} < d,
therefore X £ S^/2(\Am), that proves the first statement of Corollary.
Now let m ^ #X. Since diamX = d and 2dcn(XAm,X) = d, then, due to Theorem 7, the space X cannot be partitioned into m subsets of strictly smaller diameters.
Conversely, each X of the diameter d, such that m ^ #Iaid X cannot be partitioned into m
□
4.3. Clique Cover Number and Chromatic Number of a Simple Graph
Recall that a subgraph of an arbitrary simple graph G is called a clique, if any its two vertices are connected by an edge, i.e., the clique is a subgraph which is a complete graph itself. Notice that each single-vertex subgraph is a single-vertex clique. For convenience, the vertex set of a clique is also referred as a clique.
On the set of all cliques, an ordering with respect to inclusion is naturally defined, and hence, due to the above remarks, a family of maximal cliques is uniquely defined; this family forms a covering of the graph G in the following sense: the union of all vertex sets of all maximal cliques coincides with the vertex set V(G) of the graph G.
If one does not restrict himself by maximal cliques only, then, generally speaking, one can find other families of cliques covering the graph G. One of the classical problems of Graph Theory is to calculate the minimal possible number of cliques covering a given finite simple graph G. This number is referred as the clique cover number and is often denoted by 9(G). It is easy to see that the value 9(G) also equals the least number of cliques whose vertex sets form a partition of V(G).
Another popular problem is to find the least possible number of colors that is necessary to color the vertices of a simple finite graph G in such a way that adjacent vertices have different colors. This number is denoted by 7(G) and is referred as the chromatic number of the graph G.
For a simple graph G, by G' we denote its dual graph, i.e., the graph with the same vertex set and the complementary set of edges (two vertices of G' are adjacent if and only if they are not adjacent in G). It is not difficult to verify, that for any simple finite graph G it holds 9(G) = 7(G').
Let G = (V, E) be an arbitrary finite graph. Fix two real numbers a < b ^ 2a and define a metric on V as follows: the distance between adjacent vertices equals a, and the distance between nonadjacent vertices equals b. Then a subset V'CV has diameter a if and only if G(V')cG is a clique. This implies that the clique cover number of G equals the least possible cardinality of partitions of the metric space V into subsets of (strictly) smaller diameter. However, this number was calculated in Theorem 7. Thus, we get the following result.
Corollary 6. Let G = (V, E) be an arbitrary finite graph. Fix two real numbers a < b ^ 2a and define a metric on V as follows: the distance between adjacent vertices equals a, and the distance between nonadjacent ones equals b. Let m be the greatest positive integer k such that 2dcH(dAk, V) = b (in the case when there is no such k, we put m = 0). Then 9(G) = m + 1.
Because of the duality between clique cover and chromatic numbers, we get the following dual result.
Corollary 7. Let G = (V, E) he an arbitrary finite graph. Fix two real numbers a < b ^ 2a and define a metric on V as follows: the distance between adjacent vertices equals b, and the distance between nonadjacent ones equals a. Let m be the greatest positive integer k such that 2dch(a Ak, V) = b (in the case when there is no such k, we put m = 0). Then 7(G) = m + 1.
4.4. Examples
In conclusion we give several examples demonstrating how the above Corollaries can be applied.
4.4.1. An Empty Graph and a Complete Graph
Let G = (V,E) be an empty graph, i.e., E = 0. Put n = #V, then 9(G) = n. Now, let us calculate 9(G) by means of Corollary 6.
The metric space V constructed in Corollary 6 coincides with bAn, then for k < n we have 2da h (a A k, V) = 2da h (a A k, bAn) = b because for any R £ R(aA k, bAn) there exi sts x £ a A k such that #R(x) ^ 2, thus disE = 6. For k ^ n we have 2dcn(aAk, bAn) ^ max[a, b - a}. Indeed, for k = n we can consider a bijection R with disE = b - a. For k > n we can define R as follows: take some x £ bAn, and let E-1(^) consists of arbitrarv k - n + 1 points of aAk; for remaining points let R be a bijection. Then disE = max[a,b - a}. Thus, according to Corollary 6, we also have 9(G) = n.
Now, let G = (V,E) be a complete graph, i.e., any two its vertices are adjacent. In this case 9(G) = 1. Now, let us calculate 9(G) by means of Corollary 6.
In this case the metric space V from Corollary 6 coincides with aAn, therefore 2dah(aAk, V) = = 2dah(aAk,aAn) ^ ma^diam(aA k),diam(aAn^ < b, due to Example 4, therefore 9(G) = 1 according to Corollary 6.
4.4.2. Bipartite Graphs
Let G = (V, E) be a complete bipartite graph, i.e., its vertex set is partitioned in two non-empty non-intersecting subsets V1 and and its edge set E consists of all pairs v1v2, vi £ Vi, i = 1, 2. In this case 7(G) = 2. Now, let us calculate 7(G) by means of Corollary 7.
The metric space V constructed in Corollary 7 is a 2-distance space such that the distances between the points belonging to the same subset Vi equals a, and the distance between the points belonging to distinct Vi equals b, where 0 < a < b ^ 2a. Then diamF = 6, so doh (aA1, V) = b. Further, for aAk, k ^ 2, let us partition the vertex set of A k in two non-empty sets D1 and D2, and put R = (D1 x V-]) U (D2 x V2). Then dGh(aAk, V) ^ distE = max[a, b - a} ^ a < b. Therefore 7(G) = 2 in accordance with Corollary 7.
If G = (V,E) is a bipartite graph, i.e., its vertex set is partitioned in two non-empty non-intersecting subsets V1 and V2 again, but its edge set E is nonempty and is contained in the edge set of the corresponding complete bipartite graph, then 7(G) = 2, and similar reasoning can be used to calculate it by means of Corollary 7.
4.4.3. Distance from Simplexes to Balls and Spheres
As it is mentioned in Introduction, Lusternik and Schnirelmann [16], and a bit later independently Borsuk [14] and [15], have shown that the least possible number of parts of smaller diameter necessary to partition a sphere and a ball in Rn equals n + 1. Then Theorem 7 implies the following result.
Corollary 8. Let X be either the standard unit sphere Sn-1 or the standard unit ball Bn in the Euclidean space Rn, and 0 < A < 2. The n den (AA k ,X) < 1 for k ^ n + 1, and dc h (AA k ,X) = 1 for k ^ n.
4.4.4. Cycles and Wheel Graph
Recall that the cycle Cn with n ^ 3 vertices is a connected simple graph with n vertices and n edges, such that all the vertices have degree 2. The graphs C2k are evidently bipartite, j(C2k) = 2, and 7(C2k+i) = 3.
Let X be a finite 2-distance space with non-zero distances 0 < a < b. Construct a finite graph Gx with vertex set X connecting two vertices by an edge iff the distance between them equals b.
X
X n 2
a < b ^ 2a; such that its greater distance graph is C2k+1- Then 2dcH(aAm,X) = b for m = 1, 2.
Recall that the wheel graph Wn with n vertices is obtained from the cycle Cn-1 bv adding a single vertex and n — 1 edges connecting this vertex with all the remaining ones. It is well-known that j(W2k) = 4 and 7(W2k+i) = 3.
X n 2
a < b ^ 2a, such that its greater distance graph is Wn. If n = 2k + 1, then 2dcH(aAm, X) = b for m = 1, 2, and if n = 2k, then 2daH(aAm,X) = b for m = 1, 2, 3.
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13. J.B. Kruskal, "On the Shortest Spanning Subtree of a Graph and the Traveling Salesman Problem", Proceedings of the American Mathematical Society, 7, 48-50,1956.
14. K. Borsuk, "Uber die Zerlegung einer n-dimensionalen Vollkugel in n-Mengen". In: Ferh. International Math. Kongress Zürich, p. 192, 1932.
15. K. Borsuk, "Drei Sätze über die n-dimensionale euklidische Sphäre", Fundamenta Math., 20, 177-190, 1933.
16. Л. А. Люстерник, Л. Г. Шнирельман, Топологические методы в вариационных задачах. Исследовательский институт математики и механики при МГУ, Москва, 1930
17. G.M. Ziegler, "Colouring Hamming Graphs, Optimal Binary Codes, and the 0/1-Borsuk Problem in Low Dimensions", in: Computational Discrete Mathematics, ed. by Helmut Alt (Springer-Ver lag, Berlin Heidelberg New York, 2001), pp. 159-172.
18. H. Hadwiger, "Uberdeckung einer Menge durch Mengen kleineren Durchmessers", Commentarii Mathematici Helvetici, 18 (1), 73-75, 1945.
19. H. Hadwiger, "Mitteilung betreffend meine Note: Uberdeckung einer Menge durch Mengen kleineren Durchmessers", Commentarii Mathematici Helvetici, 19, 1946.
20. J. Kahn, G. Kalai, A counterexample to Borsuk's conjecture. Bull. Amer. Math. Soc., 29 (1), 60-62, 1993.
21. A.M. Raigorodskii, "Around Borsuk's Hypothesis", Journal of Mathematical Sciences, 154 (4), 604-623, 2008.
22. A.V. Bondarenko, "On Borsuk's conjecture for two-distance sets", arXiv e-prints, arXiv: 1305.2584, 2013.
23. T. Jenrich, "A 64-dimensional two-distance counterexample to Borsuk's conjecture", arXiv e-prints, arXiv: 1308.0206, 2013.
24. R. M.R. Lewis, A Guide to Graph Colouring, Springer, Cham, 2016.
25. A.A. Tuzhilin, Calculation of Minimum Spanning Tree Edges Lengths using Gromov-Hausdorff Distance. ArXiv e-prints, arXiv: 1605.01566, 2016.
26. А. О. Иванов, ILM. Никонов, A.A. Тужилин, "Множества, допускающие соединение графами конечной длины", Матем. сб., 196 (6), 71—110, 2005, см. также ArXiv e-prints, ArXiv:1403.383, 2014.
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22. Bondarenko, A.V. 2013, "On Borsuk's conjecture for two-distance sets", Available at: arXiv e-prints, arXiv: 1305.2584.
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24. Lewis, R. M.R. 2016, A Guide to Graph Colouring, Springer, Cham.
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