Showing papers in "Combinatorica in 1993"

Hadwiger's conjecture for K6-free graphs

Abstract: In 1943, Hadwiger made the conjecture that every loopless graph not contractible to the complete graph ont+1 vertices ist-colourable. Whent≤3 this is easy, and whent=4, Wagner's theorem of 1937 shows the conjecture to be equivalent to the four-colour conjecture (the 4CC). However, whent≥5 it has remained open. Here we show that whent=5 it is also equivalent to the 4CC. More precisely, we show (without assuming the 4CC) that every minimal counterexample to Hadwiger's conjecture whent=5 is “apex”, that is, it consists of a planar graph with one additional vertex. Consequently, the 4CC implies Hadwiger's conjecture whent=5, because it implies that apex graphs are 5-colourable.

Low diameter graph decompositions

Abstract: Adecomposition of a graphG=(V,E) is a partition of the vertex set into subsets (calledblocks). Thediameter of a decomposition is the leastd such that any two vertices belonging to the same connected component of a block are at distance ≤d. In this paper we prove (nearly best possible) statements, of the form: Anyn-vertex graph has a decomposition into a small number of blocks each having small diameter. Such decompositions provide a tool for efficiently decentralizing distributed computations. In [4] it was shown that every graph has a decomposition into at mosts(n) blocks of diameter at mosts(n) for\(s(n) = n^{O(\sqrt {\log \log n/\log n)} }\). Using a technique of Awerbuch [3] and Awerbuch and Peleg [5], we improve this result by showing that every graph has a decomposition of diameterO (logn) intoO(logn) blocks. In addition, we give a randomized distributed algorithm that produces such a decomposition and runs in timeO(log2n). The construction can be parameterized to provide decompositions that trade-off between the number of blocks and the diameter. We show that this trade-off is nearly best possible, for two families of graphs: the first consists of skeletons of certain triangulations of a simplex and the second consists of grid graphs with added diagonals. The proofs in both cases rely on basic results in combinatorial topology, Sperner's lemma for the first class and Tucker's lemma for the second.

On the complexity of the disjoint paths problem

Abstract: In this paper we consider the disjoint paths problem. Given a graphG and a subsetS of the edge-set ofG the problem is to decide whether there exists a family ℱ of disjoint circuits inG each containing exactly one edge ofS such that every edge inS belongs to a circuit inC. By a well-known theorem of P. Seymour the edge-disjoint paths problem is polynomially solvable for Eulerian planar graphsG. We show that (assumingP≠NP) one can drop neither planarity nor the Eulerian condition onG without losing polynomial time solvability. We prove theNP-completeness of the planar edge-disjoint paths problem by showing theNP-completeness of the vertex disjoint paths problem for planar graphs with maximum vertex-degree three. This disproves (assumingP≠NP) a conjecture of A. Schrijver concerning the existence of a polynomial time algorithm for the planar vertex-disjoint paths problem. Furthermore we present a counterexample to a conjecture of A. Frank. This conjecture would have implied a polynomial algorithm for the planar edge-disjoint paths problem. Moreover we derive a complete characterization of all minorclosed classes of graphs for which the disjoint paths problem is polynomially solvable. Finally we show theNP-completeness of the half-integral relaxation of the edge-disjoint paths problem. This implies an answer to the long-standing question whether the edge-disjoint paths problem is polynomially solvable for Eulerian graphs.

Simultaneous reduction of a lattice basis and its reciprocal basis

Abstract: Given a latticeL we are looking for a basisB=[b 1, ...b n ] ofL with the property that bothB and the associated basisB *=[b 1 * , ...,b * ] of the reciprocal latticeL * consist of short vectors. For any such basisB with reciprocal basisB * let $$S(B) = \mathop {\max }\limits_{1 \leqslant i \leqslant n} (|b_i | \cdot |b_i^ * |)$$ . Hastad and Lagarias [7] show that each latticeL of full rank has a basisB withS(B)≤exp(c 1·n 1/3) for a constantc 1 independent ofn. We improve this upper bound toS(B)≤exp(c 2·(lnn)2) withc 2 independent ofn. We will also introduce some new kinds of lattice basis reduction and an algorithm to compute one of them. The new algorithm proceeds by reducing the quantity $$\sum\limits_{i = 1}^n {|b|^2 } \cdot |b_i^ * |^2 $$ . In combination with an exhaustive search procedure, one obtains an algorithm to compute the shortest vector and a Korkine-Zolotarev reduced basis of a lattice that is efficient in practice for dimension up to 30.

Discrepancy and approximations for bounded VC-dimension

Abstract: Let (X, ℛ) be a set system on ann-point setX. For a two-coloring onX, itsdiscrepancy is defined as the maximum number by which the occurrences of the two colors differ in any set in ℛ. We show that if for anym-point subset $$Y \subseteq X$$ the number of distinct subsets induced by ℛ onY is bounded byO(m d) for a fixed integerd, then there is a coloring with discrepancy bounded byO(n 1/2−1/2d(logn)1+1/2d ). Also if any subcollection ofm sets of ℛ partitions the points into at mostO(m d) classes, then there is a coloring with discrepancy at mostO(n 1/2−1/2dlogn). These bounds imply improved upper bounds on the size of e-approximations for (X, ℛ). All the bounds are tight up to polylogarithmic factors in the worst case. Our results allow to generalize several results of Beck bounding the discrepancy in certain geometric settings to the case when the discrepancy is taken relative to an arbitrary measure.

A Note on Matrix Rigidity

Abstract: In this paper we give an explicit construction ofn×n matrices over finite fields which are somewhat rigid, in that if we change at mostk entries in each row, its rank remains at leastCn(log q k)/k, whereq is the size of the field andC is an absolute constant. Our matrices satisfy a somewhat stronger property, we will explain and call “strong rigidity”. We introduce and briefly discuss strong rigidity, because it is in a sense a simpler property and may be easier to use in giving explicit construction.

The influence of large coalitions

Abstract: This paper contains two results on influence in collective decision games. The first part deals with general perfect information coin-flipping games as defined in [3].Baton passing (see [8]), ann-player game from this class is shown to have the following property: IfS is a coalition of size at most $$\frac{n}{{3\log n}}$$ , then the influence ofS on the game is only $$O\left( {\frac{{\left| S \right|}}{n}} \right)$$ . This complements a result from [3] that for everyk there is a coalition of sizek with influence Ω(k/n). Thus the best possible bounds on influences of coalitions of size up to this threshold are known, and there need not be coalitions up to this size whose influence asymptotically exceeds their fraction of the population. This result may be expected to play a role in resolving the most outstanding problem in this area: Does everyn-player perfect information coin flipping game have a coalition ofo(n) players with influence 1−o(1)? (Recently Alon and Naor [1] gave a negative answer to this question.) In a recent paper Kahn, Kalai and Linial [7] showed that for everyn-variable boolean function of expectation bounded away from zero and one, there is a set of $$\frac{{n\omega (n)}}{{\log n}}$$ variables whose influence is 1−o(1), wherew(n) is any function tending to infinity withn. They raised the analogous question where 1−o(1) is replaced by any positive constant and speculated that a constant influence may be always achievable by significantly smaller sets of variables. This problem is almost completely solved in the second part of this article where we establish the existence of boolean functions where only sets of at least $$\Omega \left( {\frac{n}{{\log ^2 n}}} \right)$$ variables can have influence bounded away from zero.

Cycles through specified vertices

Abstract: Recently, various authors have obtained results about the existence of long cycles in graphs with a given minimum degreed. We extend these results to the case where only some of the vertices are known to have degree at leastd, and we want to find a cycle through as many of these vertices as possible. IfG is a graph onn vertices andW is a set ofw vertices of degree at leastd, we prove that there is a cycle through at least\(\left\lceil {\frac{w}{{\left\lceil {{n \mathord{\left/ {\vphantom {n d}} \right. \kern- ulldelimiterspace} d}} \right\rceil - 1}}} \right\rceil \) vertices ofW. We also find the extremal graphs for this property.

On the fractional matching polytope of a hypergraph

Abstract: For a hypergraph ℋ andb:ℋ→ℝ+ define Conjecture. There is a matching ℳ of ℋ such that For uniform ℋ andb constant this is the main theorem of [4]. Here we prove the conjecture if ℋ is uniform or intersecting, orb is constant.

Infinite highly arc transitive digraphs and universal covering digraphs

Abstract: A digraph (that is a directed graph) is said to be highly arc transitive if its automorphism group is transitive on the set ofs-arcs for eachs≥0. Several new constructions are given of infinite highly arc transitive digraphs. In particular, for Δ a connected, 1-arc transitive, bipartite digraph, a highly arc transitive digraphDL(Δ) is constructed and is shown to be a covering digraph for every digraph in a certain classD(Δ) of connected digraphs. Moreover, if Δ is locally finite, thenDL(Δ) is a universal covering digraph forD(Δ). Further constructions of infinite highly arc transitive digraphs are given.

The hypermetric cone is polyhedral

Abstract: The hypermetric coneH n is the cone in the spaceR n(n−1)/2 of all vectorsd=(d ij)1≤i

On graphical partitions

Abstract: An integer partition {λ1,λ2,,λ v } is said to be graphical if there exists a graph with degree sequence 〈λ i 〉 We give some results corcerning the problem of deciding whether or not almost all partitions of even integer are non-graphical We also give asymptotic estimates for the number of partitions with given rank

Fractionally colouring total graphs

Abstract: Bchzad and Vizing have conjectured that given any simple graph of maximum degree Δ, one can colour its edges and vertices with Δ+2 colours so that no two adjacent vertices, or two incident edges, or an edge and either of its ends receive the same colour. We show that for any simple graphG, V(G)ϒE(G) can be fractionally coloured with Δ+2 colours.

Boundedness of optimal matrices in extremal multigraph and digraph problems

Abstract: We consider multigraphs in which any two vertices are joined by at mostq edges, and study the Turan-type problem for a given family of forbidden multigraphs. In the caseq=2, answering a question of Brown, Erdős and Simonovits, we obtain an explicit upper bound on the size of the matrix generating an asymptotical solution of the problem. In the caseq>2 we show that some analogous statements do not hold, and so disprove a conjecture of Brown, Erdős and Simonovits.

More analysis of double hashing

Abstract: In [8], a deep and elegant analysis shows that double hashing is asymptotically equivalent to the ideal uniform hashing up to a load factor of about 0.319. In this paper we show how a randomization technique can be used to develop a surprisingly simple proof of the result that this equivalence holds for load factors arbitrarily close to 1.

Bases−cobases graphs and polytopes of matroids

Abstract: LetM be ablock matroid (i.e. a matroid whose ground setE is the disjoint union of two bases). We associate withM two objects: We prove that, ifM is graphic (or cographic), the distance between any two vertices ofG corresponding to disjoint bases is equal to the rank ofM (generalizing a result of [10]). Concerning the polytope we prove thatK is an hypercube if and only if dim(K)=rank(M). A constructive characterization of the class of matroids realizing this equality is given.

Joint extension of two theorems of Kotzig on 3-polytopes

Abstract: The weight of an edge in a graph is the sum of the degrees of its end-vertices. It is proved that in each 3-polytope there exists either an edge of weight at most 13 for which both incident faces are triangles, or an edge of weight at most 10 which is incident with a triangle, or else an edge of weight at most 8. All the bounds 13, 10, and 8 are sharp and attained independently of each other.

Correlation inequalities and a conjecture for permanents

Abstract: This paper presents conditions on nonnegative real valued functionsf 1,f 2,...,f m andg 1,g 2,...g m implying an inequality of the type $$\mathop \Pi \limits_{i = 1}^m \int {f_i (x)d\mu } (x) \leqslant \mathop \Pi \limits_{i = 1}^m \int {g_i (x)d\mu } (x).$$ This “2m-function” theorem generalizes the “4-function” theorem of [2], which in turn generalizes a “2-function” theorem ([8]) and the celebrated FKG inequality. It also contains (and was partly inspired by) an “m against 2” inequality that was deduced in [5] from a general product theorem.

Turán-Ramsey theorems and simple asymptotically extremal structures

Abstract: This paper is a continuation of [10], where P. Erdős, A. Hajnal, V. T. Sos, and E. Szemeredi investigated the following problem:

Clique coverings of the edges of a random graph

Abstract: The edges of the random graph (with the edge probabilityp=1/2) can be covered usingO(n 2lnlnn/(lnn)2) cliques. Hence this is an upper bound on the intersection number (also called clique cover number) of the random graph. A lower bound, obtained by counting arguments, is (1−ɛ)n 2/(2lgn)2.

Remarks on the density of sphere packings in three dimensions

Abstract: This paper shows how the density of sphere packings of spheres of equal radius may be studied using the Delaunay decomposition. Using this decomposition, a local notion of density for sphere packings in ℝ3 is defined. Conjecturally this approach should yield a bound of 0.740873... on sphere packings in ℝ3, and a small perturbation of this approach should yield the bound of $${\pi \mathord{\left/ {\vphantom {\pi {\sqrt {18} }}} \right. \kern- ulldelimiterspace} {\sqrt {18} }}$$ . The face-centered-cubic and hexagonal-close-packings provide local maxima (in a strong sense defined below) to the function which associates to every saturated sphere packing in ℝ3 its density. The local measure of density coincides with the actual density for the face-centered cubic and hexagonal-close-packings.

Homomorphisms to oriented cycles

Abstract: We discuss the existence of homomorphisms to oriented cycles and give, for a special class of cyclesC, a characterization of those digraphs that admit, a homomorphism toC. Our result can be used to prove the multiplicativity of a certain class of oriented cycles, (and thus complete the characterization of multiplicative oriented cycles), as well as to prove the membership of the corresponding decision problem in the classNPϒcoNP. We also mention a conjecture on the existence of homomorphisms to arbitrary oriented cycles.

A generalization of the AZ identity

Abstract: The identity discovered in [1] can be viewed as a sharpening of the LYM inequality ([3], [4], [5]). It was extended in [2] so that it covers also Bollobas' inequality [6]. Here we present a further generalization and demonstrate that it shares with its predecessors the usefullness for uniqueness proofs in extremal set theory.

On kernels in perfect graphs

Abstract: A kernel of a digraphD is a set of vertices which is both independent and absorbant. In 1983, C. Berge and P. Duchet conjectured that an undirected graphG is perfect if and only if the following condition is fulfilled: ifD is an orientation ofG (where pairs of opposite arcs are allowed) and if every clique ofD has a kernel thenD has a kernel. We prove here the conjecture for the complements of strongly perfect graphs and establish that a minimal counterexample to the conjecture is not a complete join of an independent set with another graph.

Decomposition of 3-connected graphs

Abstract: Cunningham and Edmonds [4[ have proved that a 2-connected graphG has a unique minimal decomposition into graphs, each of which is either 3-connected, a bond or a polygon. They define the notion of a good split, and first prove thatG has a unique minimal decomposition into graphs, none of which has a good split, and second prove that the graphs that do not have a good split are precisely 3-connected graphs, bonds and polygons. This paper provides an analogue of the first result above for 3-connected graphs, and an analogue of the second for minimally 3-connected graphs. Following the basic strategy of Cunningham and Edmonds, an appropriate notion of good split is defined. The first main result is that ifG is a 3-connected graph, thenG has a unique minimal decomposition into graphs, none of which has a good split. The second main result is that the minimally 3-connected graphs that do not have a good split are precisely cyclically 4-connected graphs, twirls (K3,n for somen≥3) and wheels. From this it is shown that ifG is a minimally 3-connected graph, thenG has a unique minimal decomposition into graphs, each of which is either cyclically 4-connected, a twirl or a wheel.

k-sets and random hulls

Abstract: We re-examine the probabilistic analysis of Clarkson and Shor [5] involvingk-sets of point sets and related structures. By studying more carefully the equations that they derive, we are able to obtain refined analysis of these quantities, which lead to a collection of interesting relationships involvingk-sets, convex hulls of random samples, and generalizations of these constructs.

A note on arrays of dots with distinct slopes

Abstract: We prove that the maximum number of dots in ann×n array of dots with distinct slopes is at leastcn 2/3(logn)−1/3 withc>0. This improves a previous result ofcn 1/2. An upper bound isO(n 4/5).

Separation of a finite set in R d by spanned hyperplanes.

Abstract: A question of the following kind will concern us here: what is the minimal numbern, ensuring that any spanning set ofn points in 3-space spans a plane, every open side of which contains at least, say, 1000 points of the set. The answer isn=4001 (see Theorem 2.1 below).

Euclideaness and final polynomials in oriented matroid theory

Abstract: This paper deals with a geometric construction of algebraic non-realizability proofs for certain oriented matroids. As main result we obtain an algorithm which generates a (bi-quadratic) final polynomial [3], [5] for any non-euclidean oriented matroid. Here we apply the results of Edmonds, Fukuda and Mandel [6], [7] concerning non-degenerate cycling of linear programs in non-euclidean oriented matroids.