Showing papers in "Combinatorica in 1984"
TL;DR: It is proved that given a polytopeP and a strictly interior point a εP, there is a projective transformation of the space that mapsP, a toP′, a′ having the following property: the ratio of the radius of the smallest sphere with center a′, containingP′ to theradius of the largest sphere withCenter a′ contained inP′ isO(n).
Abstract: We present a new polynomial-time algorithm for linear programming. In the worst case, the algorithm requiresO(n
3.5
L) arithmetic operations onO(L) bit numbers, wheren is the number of variables andL is the number of bits in the input. The running-time of this algorithm is better than the ellipsoid algorithm by a factor ofO(n
2.5). We prove that given a polytopeP and a strictly interior point a eP, there is a projective transformation of the space that mapsP, a toP′, a′ having the following property. The ratio of the radius of the smallest sphere with center a′, containingP′ to the radius of the largest sphere with center a′ contained inP′ isO(n). The algorithm consists of repeated application of such projective transformations each followed by optimization over an inscribed sphere to create a sequence of points which converges to the optimal solution in polynomial time.
4,806 citations
TL;DR: The aim of this paper is to show that the minimum Hadwiger number of graphs with average degreek is O(k/√logk), and it follows that hadwiger’s conjecture is true for almost all graphs withn vertices andnk edges.
Abstract: The aim of this paper is to show that the minimum Hadwiger number of graphs with average degreek isO(k/√logk). Specially, it follows that Hadwiger’s conjecture is true for almost all graphs withn vertices, furthermore ifk is large enough then for almost all graphs withn vertices andnk edges.
358 citations
TL;DR: The bound (t+1)(k-t-1) represents the best possible strengthening of the original 1961 theorem of Erdös, Ko, and Rado which reaches the same conclusion under the hypothesisn≧t+(k−t) .
Abstract: This paper contains a proof of the following result: ifn≧(t+1)(k−t−1), then any family ofk-subsets of ann-set with the property that any two of the subsets meet in at leastt points contains at most\(\left( {\begin{array}{*{20}c} {n - t} \\ {k - t} \\ \end{array} } \right)\) subsets. (By a theorem of P. Frankl, this was known whent≧15.) The bound (t+1)(k-t-1) represents the best possible strengthening of the original 1961 theorem of Erdos, Ko, and Rado which reaches the same conclusion under the hypothesisn≧t+(k−t)\(\left( {\begin{array}{*{20}c} k \\ t \\ \end{array} } \right)^3 \). Our proof is linear algebraic in nature; it may be considered as an application of Delsarte’s linear programming bound, but somewhat lengthy calculations are required to reach the stated result. (A. Schrijver has previously noticed the relevance of these methods.) Our exposition is self-contained.
314 citations
[...]
TL;DR: Givenn random red points on the unit square, the transportation cost between them is tipically √n logn, where logn is the number of red points in the ellipsoidal plane.
Abstract: Givenn random red points on the unit square, the transportation cost between them is tipically √n logn.
273 citations
TL;DR: The truth of Karp's conjecture is shown to follow from another conjecture concerning group actions on topological spaces and a special case of the conjecture is proved which is applied to prove Karp’s conjecture for the case of properties of graphs on a prime power number of vertices.
Abstract: The complexity of a digraph property is the number of entries of the vertex adjacency matrix of a digraph which must be examined in worst case to determine whether the graph has the property Rivest and Vuillemin proved the result (conjectured by Aanderaa and Rosenberg) that every graph property that is monotone (preserved by addition of edges) and nontrivial (holds for some but not all graphs) has complexity Ω(v
2) wherev is the number of vertices Karp conjectured that every such property is evasive, ie, requires that every entry of the incidence matrix be examined In this paper the truth of Karp’s conjecture is shown to follow from another conjecture concerning group actions on topological spaces A special case of the conjecture is proved which is applied to prove Karp’s conjecture for the case of properties of graphs on a prime power number of vertices
165 citations
TL;DR: A negative answer is given by showing that 1 - \frac{1}{{l^{r - 1} }} is not a jump ifr≧3,l>2r, and edges, wherec=c(α) does not depend on ε andm.
Abstract: The number α, 0≦α≦1, is a jump forr if for any positive e and any integerm,m≧r, anyr-uniform hypergraph withn>n
o (e,m) vertices and at least (α+e)
$$\left( {\begin{array}{*{20}c} n \\ r \\ \end{array} } \right)$$
edges contains a subgraph withm vertices and at least (α+c)
$$\left( {\begin{array}{*{20}c} m \\ r \\ \end{array} } \right)$$
edges, wherec=c(α) does not depend on e andm. It follows from a theorem of Erdos, Stone and Simonovits that forr=2 every α is a jump. Erdos asked whether the same is true forr≧3. He offered $ 1000 for answering this question. In this paper we give a negative answer by showing that
$$1 - \frac{1}{{l^{r - 1} }}$$
is not a jump ifr≧3,l>2r.
149 citations
TL;DR: A simple algorithm is developed which allows to decide whether or not a given hexagonal system has a perfect matching (and to find such a matching).
Abstract: A simple algorithm is developed which allows to decide whether or not a given hexagonal system has a perfect matching (and to find such a matching). This decision is also of chemical relevance since a hexagonal system is the skeleton of a benzenoid hydrocarbon molecule if and only if it has a perfect matching.
99 citations
TL;DR: Which colourings of the subgraphs isomorphic to H in Kn must always occur (and which types of colourings can occur when Kn is coloured in an appropriate way)?
Abstract: Given a sample graphH and two integers,n andr, we colourK
n
byr colours and are interested in the following problem. Which colourings of the subgraphs isomorphic to H in K
n
must always occur (and which types of colourings can occur whenK
n
is coloured in an appropriate way)? These types of problems include theRamsey theory, where we ask: for whichn andr must a monochromaticH occur. They also include theanti-Ramsey type problems, where we are trying to ensure a totally multicoloured copy ofH, that is, anH each edge of which has different colour.
67 citations
TL;DR: It is proved that the edge set of an arbitrary simple graphG onn vertices can be covered by at mostn−[log2n]+1 complete bipartite subgraphs of G if the weight of a subgraph is the number of its vertices and there always exists a cover with total weightc(n2/logn).
Abstract: We prove that the edge set of an arbitrary simple graphG onn vertices can be covered by at mostn−[log2
n]+1 complete bipartite subgraphs ofG. If the weight of a subgraph is the number of its vertices, then there always exists a cover with total weightc(n
2/logn) and this bound is sharp apart from a constant factor. Our result answers a problem of T. G. Tarjan.
67 citations
TL;DR: An error in the proof, and in the statement of a generalization, of the result that submodular setfunctions can be minimized over the subsets with odd cardinality is corrected.
Abstract: An error in the proof, and in the statement of a generalization, of the result that submodular setfunctions can be minimized over the subsets with odd cardinality is corrected.
63 citations
TL;DR: For every integerd>2 the authors give an explicit construction of infinitely many Cayley graphsX of degreed withn(X) vertices and girth >0.4801...(logn(X))/log (d−1)−2.
Abstract: For every integerd>2 we give an explicit construction of infinitely many Cayley graphsX of degreed withn(X) vertices and girth >0.4801...(logn(X))/log (d−1)−2. This improves a result of Margulis.
TL;DR: An explicit construction of Biggs and Hoare yields an infinite family of bipartite cubic graphs and it is proved that the ordern and girthg of each of these graphs are related by log2n<3/4·g+3/2.
Abstract: An explicit construction of Biggs and Hoare yields an infinite family of bipartite cubic graphs. We prove that the ordern and girthg of each of these graphs are related by log2
n<3/4·g+3/2.
TL;DR: Conditions are found under which the expected number of automorphisms of a large random labelled graph with a given degree sequence is close to 1, which produces an asymptotic formula for the number of unlabelledk-regular simple graphs onn vertices, as well as various asymPTotic results on the probable connectivity and girth of such graphs.
Abstract: Conditions are found under which the expected number of automorphisms of a large random labelled graph with a given degree sequence is close to 1. These conditions involve the probability that such a graph has a given subgraph. One implication is that the probability that a random unlabelledk-regular simple graph onn vertices has only the trivial group of automorphisms is asymptotic to 1 asn → ∞ with 3≦k=O(n
1/2−c). In combination with previously known results, this produces an asymptotic formula for the number of unlabelledk-regular simple graphs onn vertices, as well as various asymptotic results on the probable connectivity and girth of such graphs. Corresponding results for graphs with more arbitrary degree sequences are obtained. The main results apply equally well to graphs in which multiple edges and loops are permitted, and also to bicoloured graphs.
TL;DR: In this chapter the exact value off(r, 3) is determined and (asymptotically sharp) bounds onf( r, k),k>3 are established and an analogue of Sauer’ theorem for uniform set-systems is proved.
Abstract: A system of setsE
1,E
2, ...,E
k ⊂X is said to be disjointly representable if there existx
1,x
2, ...,x
k
teX such thatx
i
teE
j
⇔i=j. Letf(r, k) denote the maximal size of anr-uniform set-system containing nok disjointly representable members. In the first section the exact value off(r, 3) is determined and (asymptotically sharp) bounds onf(r, k),k>3 are established. The last two sections contain some generalizations, in particular we prove an analogue of Sauer’ theorem [16] for uniform set-systems.
TL;DR: It is proved that the extreme points of this set in Rn+1 have at most two non-zero components.
Abstract: Let ℱ be a family of subsets of a finite set ofn elements. The vector (f
0, ...,f
n
) is called the profile of ℱ wheref
i
denotes the number ofi-element subsets in ℱ. Take the set of profiles of all families ℱ satisfyingF
1⊄F
2 andF
1∩F
2≠0 for allF
1,F
2teℱ. It is proved that the extreme points of this set inR
n+1 have at most two non-zero components.
TL;DR: It is proved that fork≧2 (0, …, 0) and (i=1, ...,n) are the vertices of 〈ℓ〉, the convex closure in the (n+1)-dimensional Euclidean space.
Abstract: LetP be that partially ordered set whose elements are vectors x=(x
1, ...,x
n
) withx
i
e {0, ...,k} (i=1, ...,n) and in which the order is given byx≦y iffx
i
=y
i
orx
i
=0 for alli. LetN
i
(P)={x eP : |{j:x
j
≠ 0}|=i}. A subsetF ofP is called an Erdos-Ko-Rado family, if for allx, y eF it holdsx ≮y, x ≯ y, and there exists az eN
1(P) such thatz≦x andz≦y. Let l be the set of all vectorsf=(f
0, ...,f
n
) for which there is an Erdos-Ko-Rado familyF inP such that |N
i
(P) ∩F|=f
i
(i=0, ...,n) and let 〈l〉 be its convex closure in the (n+1)-dimensional Euclidean space. It is proved that fork≧2 (0, ..., 0) and
$$\left( {0,...,0,\overbrace {i - component}^{\left( {\begin{array}{*{20}c} {n - 1} \\ {i - 1} \\ \end{array} } \right)}k^{i - 1} ,0,...,0} \right)$$
(i=1, ...,n) are the vertices of 〈l〉.
TL;DR: It is proved that if |ℓ|>3.5 then ℓ contains four distinct members A, B, C, D such that A∪B=C∪D and A∩B =C∩D=0.
Abstract: Let l be a set-system ofr-element subsets on ann-element set,r≧3. It is proved that if |l|>3.5\(\left( {\begin{array}{*{20}c} n \\ {r - 1} \\ \end{array} } \right)\) then l contains four distinct membersA, B, C, D such thatA∪B=C∪D andA∩B=C∩D=0.
TL;DR: It was shown before that ifG is a graph of maximum degreep containing no cliques of the sizeq then the independence ratio is greater than or equal to 2 / (p +q).
Abstract: It was shown before that ifG is a graph of maximum degreep containing no cliques of the sizeq then the independence ratio is greater than or equal to 2 / (p +q). We shall discuss here some extreme cases of this inequality.
TL;DR: It is established that decidingt-colorability for a simplek-graph whent≧3,k ≤3 is NP-complete and it is shown that if there is a polynomial time algorithm for finding the chromatic number of a Steiner Triple system then such an algorithm would imply that P=NP.
Abstract: In this paper we establish that decidingt-colorability for a simplek-graph whent≧3,k≧3 is NP-complete. Next, we establish that if there is a polynomial time algorithm for finding the chromatic number of a Steiner Triple system then there exists a polynomial time “approximation” algorithm for the chromatic number of simple 3-graphs. Finally, we show that the existence of such an approximation algorithm would imply that P=NP.
TL;DR: Two graphs are defined which are very similar and it is shown thatΓ is contained in every graph of uncountable chromatic number, whileΔ is (consistently) not.
Abstract: We investigate the following problem: What countable graphs must a graph of uncountable chromatic number contain? We define two graphsΓ andΔ which are very similar and we show thatΓ is contained in every graph of uncountable chromatic number, whileΔ is (consistently) not.
TL;DR: The difference between the notions of the ordinary and measurable chromatic numbers is investigated and upper and lower bounds on the Lebesgue upper density of independent sets are proved.
Abstract: LetH be a set of positive real numbers. We define the geometric graphG
H
as follows: the vertex set isR
n
(or the unit circleS
1) andx, y are joined if their distance belongs toH. We define the measurable chromatic number of geometric graphs as the minimum number of classes in a measurable partition into independent sets. In this paper we investigate the difference between the notions of the ordinary and measurable chromatic numbers. We also prove upper and lower bounds on the Lebesgue upper density of independent sets.
TL;DR: This work studies equidistant codes of length 4k + 1 having (constant) weight 2k, and ( constant) distance 2k between codewords, and can construct such a code, with 4k − 1 codeword, from a (4k − 2, 2k −1, k − 1) — SBIBD.
Abstract: We study equidistant codes of length 4k + 1 having (constant) weight 2k, and (constant) distance 2k between codewords. The maximum number of codewords is 4k; this can be attained if and only ifk = (u
2 +u)/2 (for some integeru) and there exists a ((2u
2 + 2u + 1,u
2, (u
2 −u)/2) — SBIBD. Also, one can construct such a code, with 4k − 1 codewords, from a (4k − 1, 2k − 1,k − 1) — SBIBD.
TL;DR: This paper proves a stronger result which provides an upper bound on the chromatic number of a graph in which it is shown that the graph onn vertices has no odd cycle with length less thanckn1/k, and also proves that the number of subgraphs with small diameter is bound.
Abstract: In 1973, P. Erdos conjectured that for eachke2, there exists a constantc
k
so that ifG is a graph onn vertices andG has no odd cycle with length less thanc
k
n
1/k
, then the chromatic number ofG is at mostk+1. Constructions due to Lovasz and Schriver show thatc
k
, if it exists, must be at least 1. In this paper we settle Erdos’ conjecture in the affirmative. We actually prove a stronger result which provides an upper bound on the chromatic number of a graph in which we have a bound on the chromatic number of subgraphs with small diameter.
TL;DR: It is shown that if an edge is removed from a 3-connected cubic graph, then there is still a cycle through any five vertices, and the circumstances in which there fails to be a cycleThrough six are characterised.
Abstract: It is known that there exists a cycle through any nine vertices of a 3-connected cubic graphG. Here we show that if an edge is removed from such a graph, then there is still a cycle through any five vertices. Furthermore, we characterise the circumstances in which there fails to be a cycle through six. As corollaries we are able to prove that a 3-connected cubic graph has a cycle through any specified five vertices and one edge, and to classify the conditions under which it has a cycle through four chosen vertices and two edges. We are able to use the five and six vertex results to show that a 3-connected cubic graph has a cycle which passes through any ten given vertices if and only if the graph is not contractible to the Petersen graph in such a way that the ten vertices each map to a distinct vertex of the Petersen graph.
TL;DR: This work determines the asymptotic behaviour of various parameters associated with unlabelled directed graphs in which each point has out-degree one, such as the expected number of points belonging to cycles and theexpected number of components.
Abstract: Random mapping patterns may be represented by unlabelled directed graphs in which each point has out-degree one. We determine the asymptotic behaviour of various parameters associated with such graphs, such as the expected number of points belonging to cycles and the expected number of components.
TL;DR: This paper determines precisely when a matroidM has a quad, a 4-element set which is the intersection of a circuit and a cocircuit, and shows that this will occur if M has a Circuit and a Cocircuit meeting in more than four elements.
Abstract: Seymour has shown that a matroid has a triad, that is, a 3-element set which is the intersection of a circuit and a cocircuit, if and only if it is non-binary In this paper we determine precisely when a matroidM has a quad, a 4-element set which is the intersection of a circuit and a cocircuit We also show that this will occur ifM has a circuit and a cocircuit meeting in more than four elements In addition, we prove that if a 3-connected matroid has a quad, then every pair of elements is in a quad The corresponding result for triads was proved by Seymour
[...]
TL;DR: It is found that there are examples ofH such that copies ofG do not form short cycles in H, providing a strenghtening of the following well-known result of Erdős: there exist graphs with high chromatic number and no short cycles.
Abstract: IfH is a Ramsey graph for a graphG thenH is rich in copies of the graphG. Here we prove theorems in the opposite direction. We find examples ofH such that copies ofG do not form short cycles inH. This provides a strenghtening also, of the following well-known result of Erdős: there exist graphs with high chromatic number and no short cycles. In particular, we solve a problem of J. Spencer.
TL;DR: It is shown thatGeodetic blocks of diameter 3 are self-centred and upper and lower geodetic critical and also lower diameter critical.
Abstract: It is shown that geodetic blocks of diameter 3 are self-centred and upper and lower geodetic critical and also lower diameter critical. Geodetic blocks of diameter 3 which are isomorphic toK n (2) are characterised.
TL;DR: It is shown that if a treeS is bisectable then there is a unique treeT that accomplishes the bipartition and with the help of the uniqueness ofT and the observation that the bisection ofS into two copies ofT is unique up to isomorphism, this result is contained in the proof of a theorem.
Abstract: A tree is called even if its line set can be partitioned into two isomorphic subforests; it is bisectable if these forests are trees. The problem of deciding whether a given tree is even is known (Graham and Robinson) to be NP-hard. That for bisectability is now shown to have a polynomial time algorithm. This result is contained in the proof of a theorem which shows that if a treeS is bisectable then there is a unique treeT that accomplishes the bipartition. With the help of the uniqueness ofT and the observation that the bisection ofS into two copies ofT is unique up to isomorphism, we enumerate bisectable trees.
TL;DR: This paper investigates the case |L|=3, and shows that in some cases m(n, k, L)=O(n3/2), which is quite surprising.
Abstract: LetX be a finite set ofn elements and l a family ofk-subsets ofX. Suppose that for a given setL of non-negative integers all the pairwise intersections of members of l have cardinality belonging toL. Letm(n, k, L) denote the maximum possible cardinality of l. This function was investigated by many authors, but to determine its exact value or even its correct order of magnitude appears to be hopeless. In this paper we investigate the case |L|=3. We give necessary and sufficient conditions form(n, k, L)=O(n) andm(n, k, L)≧O(n
2), and show that in some casesm(n, k, L)=O(n
3/2), which is quite surprising.