Showing papers in "Graphs and Combinatorics in 1985"
TL;DR: It is shown that ifc>11/6 and Δ is sufficiently large, then a graph of maximum degree Δ with a list of cardinality [cΔ] assigned to each edge may be edge-coloured so that each edge is coloured with an element of its list.
Abstract: In this paper, we go some way towards proving a conjecture of Albertson and Tucker. Among other results, we show that ifc>11/6 and Δ is sufficiently large, then a graph of maximum degree Δ with a list of cardinality [cΔ] assigned to each edge may be edge-coloured so that each edge is coloured with an element of its list.
146 citations
TL;DR: A combinatorial interpretation ofSI a product of homogeneous symmetric functions andJ, K unrestricted skew shapes is given and how Gessel's and Lascoux's results are related is shown and shown how they can be derived from a special case of the result.
Abstract: The Kronecker product of two homogeneous symmetric polynomialsP 1,P 2 is defined by means of the Frobenius map by the formulaP 1oP 2=F(F ?1 P 1)(F ?1 P 2). WhenP 1 andP 2 are the Schur functionsS I ,S J then the resulting productS I oS J is the Frobenius characteristic of the tensor product of the two representations corresponding to the diagramsI andJ. Taking the scalar product ofS I oS J with a third Schur functionsS K gives the so called Kronecker coefficientc I,J,K = . In recent work lascoux [7] and Gessel [3] have given what appear to be two separate combinatorial interpretations for thec I,J,K in terms of some classes of permutations. In Lascoux's workI andJ are restricted to be hooks and in Gessel's both have to be zigzag partitions. In this paper we give a general result relating shuffles of permutations and Kronecker products. This leads us to a combinatorial interpretation of forS I a product of homogeneous symmetric functions andJ, K unrestricted skew shapes. We also show how Gessel's and Lascoux's results are related and show how they can be derived from a special case of our result.
100 citations
TL;DR: All symmetric designs are determined for which the automorphism group is 2-transitive on the set of points.
Abstract: All symmetric designs are determined for which the automorphism group is 2-transitive on the set of points.
81 citations
TL;DR: This paper establishes the following generalization which was suggested by Erdös: for each positive constantc and positive integerk there exist positive integersfk(c) andno such that ifG is any graph with more thanno vertices having the property that the chromatic number ofG cannot be made less thank by the omission of at mostcn2 edges, thenG contains ak-chromatic subgraph with at mostfk (c) vertices.
Abstract: Bollobas, Erdos, Simonovits, and Szemeredi conjectured [1] that for each positive constantc there exists a constantg(c) such that ifG is any graph which cannot be made 3-chromatic by the omission ofcn 2 edges, thenG contains a 4-chromatic subgraph with at mostg(c) vertices. Here we establish the following generalization which was suggested by Erdos [2]: For each positive constantc and positive integerk there exist positive integersf k(c) andn o such that ifG is any graph with more thann o vertices having the property that the chromatic number ofG cannot be made less thank by the omission of at mostcn 2 edges, thenG contains ak-chromatic subgraph with at mostf k(c) vertices.
78 citations
54 citations
TL;DR: Criterions for the existence of a {1,3,...,2n−1}-factor in a tree and in a graph are given.
Abstract: A {1,3,...,2n?1}-factor of a graphG is defined to be a spanning subgraph ofG each degree of whose vertices is one of {1,3,...,2n?1}, wheren is a positive integer. In this paper, we give criterions for the existence of a {1,3,...,2n?1}-factor in a tree and in a graph.
40 citations
TL;DR: This theorem generalizes a theorem which gives a solution for the problem without the condition and gives a necessary and sufficient condition for the uniqueness of the solution in the KKL theorem.
Abstract: LetA be a set ofk-subsets of {1,...,n} with[Figure not available: see fulltext.]. We determine the minimal cardinality of thel-shadow ofA. This theorem generalizes a theorem of Kruskal [10], Katona [8], Lindstrom [11] and other which gives a solution for the problem without the condition[Figure not available: see fulltext.]. Furthermore we give a necessary and sufficient condition for the uniqueness of the solution in the KKL theorem.
40 citations
TL;DR: It is proved that there exists a Steiner triple system of orderv that can be nested if and only if vэ1 mod 6 is obtained.
Abstract: A Steiner triple system can benested if it is possible to add one point to each block in such a way that a BIBD with block-size 4 and ?=1 is obtained. We prove that there exists a Steiner triple system of orderv that can be nested if and only ifv?1 mod 6.
40 citations
TL;DR: The existence of Hadamard matrices of order 268 is established and the existence of Baumert-Hall arrays of order 335, and 603 is established as well.
Abstract: The existence of Hadamard matrices of order 268 is established. More generally, suppose that there exist Williamson matrices of orderr. It is shown that this implies the existence of a Hadamard matrix of order 268r. The existence of Baumert-Hall arrays of order 335, and 603 is established as well.
39 citations
TL;DR: It is proved that Kn has an edgecoloring admitting no antihomogeneous subset of size 8 (nlnn)1/3, answering a question of V. Rödl.
Abstract: An edge-coloring of a graph is a partition of the set of edges into color-classes such that no two edges in the same class are adjacent. A subsetA of the vertex set isantihomogeneous if all edges in the subgraph induced byA have different colors. We study the maximum size of antihomogeneous subsets of the complete graphK n with respect to edge-colorings. The greedy algorithm shows that this size is always at least (2n)1/3. On the other hand, we prove thatK n has an edgecoloring admitting no antihomogeneous subset of size 8 (nlnn)1/3, answering a question of V. Rodl. The proof is not constructive; it employs a probabilistic argument. Related problems concerning Sidon sets in groups are mentioned.
36 citations
TL;DR: An equation is derived which is satisfied by special types of generating functions for labelled chordal graphs with given numbers of cliques of given sizes, and the number of n-vertex labelled chordAL graphs withgiven connectivity is determined.
Abstract: An equation is derived which is satisfied by special types of generating functions for labelled chordal graphs. This enables calculation of the numbers of labelled chordal graphs with given numbers of cliques of given sizes. From this is determined the number ofn-vertex labelled chordal graphs with given connectivity. Calculations were completed forn≤13.
TL;DR: It is proved that fora≥1 fixed andt sufficiently largeT(n, t+a,t)>(1-a(a+4+o(1))logt/(at)(tn holds).
Abstract: Turan's problem is to determine the maximum numberT(n,k,t) oft-element subsets of ann-set without a complete sub-hypergraph onk vertices, i.e., allt-subsets of ak-set. It is proved that fora?1 fixed andt sufficiently largeT(n, t+a,t)>(1-a(a+4+o(1))logt/( a t )( t n holds
TL;DR: In this article, a necessary and sufficient condition for a hexagonal system to have a perfect matching is obtained, based on the Sachs's conjecture, and the condition is shown to hold for all hexagonal systems.
Abstract: This paper deals with perfect matchings in hexagonal systems. Counterexamples are given to Sachs's conjecture in this field. A necessary and sufficient condition for a hexagonal system to have a perfect matching is obtained.
TL;DR: In this article, the validity of Ivan Gutman's conjecture concerning covers of hexagonal systems was established, and the conjecture was shown to hold for the covers of a hexagonal system.
Abstract: In this paper we establish the validity of Ivan Gutman's conjecture concerning covers of hexagonal systems.
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TL;DR: In order to provide an algebraic graph theoretic background for Hadamard matrices and designs hadamard graphs are introduced and their spectra are determined.
Abstract: In order to provide an algebraic graph theoretic background for Hadamard matrices and designs Hadamard graphs are introduced and their spectra are determined.
TL;DR: This work pursues the technique of “holes” to study the retracts of an ordered set to establish a close connection between theclass of absolute retracts and the class of dismantlable ordered sets.
Abstract: We pursue the technique of "holes" to study the retracts of an ordered set. This is applied to establish a close connection between the class of absolute retracts and the class of dismantlable ordered sets.
TL;DR: A formula for the splitting number of the complete graph is derived by virtue of vertex identifications from a suitable planar graph.
Abstract: If a given graphG can be obtained bys vertex identifications from a suitable planar graph ands is the minimum number for which this is possible thens is called the splitting number ofG. Here a formula for the splitting number of the complete graph is derived.
TL;DR: The number of disjoint pairs in ℱ is bounded by (1+o(1))22n, which proves an old conjecture of Erdös-Stone type and verifies a conjecture of Daykin and Erd Ös.
Abstract: Let ? be a family of 2 n+1 subsets of a 2n-element set. Then the number of disjoint pairs in ? is bounded by (1+o(1))22n . This proves an old conjecture of Erdos. Let ? be a family of 21/(k+1)+?)n subsets of ann-element set. Then the number of containments in ? is bounded by (1-1/k+o(1))( 2 |?| ). This verifies a conjecture of Daykin and Erdos. A similar Erdos-Stone type result is proved for the maximum number of disjoint pairs in a family of subsets.
TL;DR: In graphs, for every pair of vertices x andy in a connected, finite, undirected graphG, there is a pathP joiningx andy such that deleting the edges ofP fromG reduces the local edge-connectivity by at most two.
Abstract: For every pair of verticesx andy in a connected, finite, undirected graphG, there is a pathP joiningx andy such that deleting the edges ofP fromG, for every pair of vertices ofG, the local edge-connectivity decreases by at most two.
TL;DR: It is shown that for every graphG=(V,E) and every initial signs, there is a sequencev1,v2,...,vr of vertices of G, in which no vertex appears more than once, such that ifvi becomes active at timei, (1≤i≤r), then after theser stepsG reaches a stable state.
Abstract: LetG=(V,E) be a graph with an initial signs(v)?{±1} for every vertexv?V. When a certexv becomesactive, it resets its sign tos?(v) which is the sign of the majority of its neighbors(s?(v)=1 if there is a tie).G is in astable state if,s?(v) for allv?V. We show that for every graphG=(V,E) and every initial signs, there is a sequencev 1,v 2,...,v r of vertices ofG, in which no vertex appears more than once, such that ifv i becomes active at timei, (1≤i≤r), then after theser stepsG reaches a stable state. This proves a conjecture of Miller. We also consider some generalizations to directed graphs with weighted edges.
TL;DR: Cyclic automorphisms of the countable universal ultrahomogeneous graph are investigated and it is shown that the probability that such a set consists entirely of odd numbers is strictly positive.
Abstract: Cyclic automorphisms of the countable universal ultrahomogeneous graph are investigated using methods of Baire category and measure theory. This leads to the study of random sumfree sets; it is shown that the probability that such a set consists entirely of odd numbers is strictly positive, and bounds are given.
TL;DR: The smallest dimensions are determined for the join of a large complete graph and an empty graph, and for complete multipartite graphs with more vertex classes than the size of its largest vertex class.
Abstract: LetF be a set of nonoverlapping spheres in Euclideann-spaceE n . By the contact pattern ofF we mean the graph whose vertex set isF and two vertices are adjacent whenever the corresponding spheres touch each other. Every graph turns out to be a contact pattern in some dimension. This paper studies the smallest dimensionn for a graphG such thatG is a contact pattern inE n . Among others, the smallest dimensions are determined for the join of a large complete graph and an empty graph, and for complete multipartite graphs with more vertex classes than the size of its largest vertex class.
TL;DR: A generalization of Hall's theorem is proved for fractional matchings inn-partiten-regular hypergraphs.
Abstract: A generalization of Hall's theorem is proved for fractional matchings inn-partiten-regular hypergraphs.
TL;DR: If such a graph is cyclically 4-edge-connected with order greater than 8 it is shown that any four independent edges lie on a cycle.
Abstract: We give necessary and sufficient conditions for four edges in a 3-connected cubic graph to lie on a cycle. As a consequence, if such a graph is cyclically 4-edge-connected with order greater than 8 it is shown that any four independent edges lie on a cycle.
TL;DR: Some properties of a graph with bind(G)=(|V (G)|−1)/(|V(G) |−ρ(G)) are given, and the binding number of some line graphs and total graphs are determined.
Abstract: D.R. Woodall [7] introduced the concept of the binding number of a graphG, bind (G), and proved that bind(G)?(|V(G)|?1)/(|V(G)|??(G)). In this paper, some properties of a graph with bind(G)=(|V(G)|?1)/(|V(G)|??(G)) are given, and the binding number of some line graphs and total graphs are determined.
TL;DR: Solving a problem of Erdös the existence of a graph withn vertices,cn2 edges and withoutK(4) andK(3, 3, 3) is proved.
Abstract: Solving a problem of Erdos the existence of a graph withn vertices,cn 2 edges and withoutK(4) andK(3, 3, 3) is proved. It remains an open question whether all such graphs containK(2, 2, 2).
TL;DR: It is shown that the extendability corresponds to the existence of a proper family of maximal arcs, which implies a natural duality between point and block arcs is established, which among other things implies a result of Cameron and van Lint that extendability of a given design in this family is equivalent to extendable of its dual.
Abstract: An (?,n)-arc in a 2-design is a set ofn points of the design such that any block intersects it in at most ? points. For such an arc,n is bounded by 1+(r(??1)/?), with equality if and only if every block meets the arc in either 0 or ? points. An (?,n) arc with equality in above is said to be maximal.
A maximal block arc can be dually defined. This generalizes the notion of an oval (?=2) in a symmetric design due to Asmus and van Lint. The aim of this paper is to study the infinite family of possibly extendable symmetric designs other than the Hadamard design family and their related designs using maximal arcs. It is shown that the extendability corresponds to the existence of a proper family of maximal arcs. A natural duality between point and block arcs is established, which among other things implies a result of Cameron and van Lint that extendability of a given design in this family is equivalent to extendability of its dual. Similar results are proved for other related designs.
TL;DR: A scheme for addressing digraphs is proposed, and the minimum address length is studied both in general and in certain special cases.
Abstract: The "distance" from vertexu to vertexv in a strongly connected digraph is the number of arcs in a shortest directed path fromu tov. The addressing problem, first formulated in the undirected case by Graham and Pollak, entails the assignment of a string of symbols to each vertex in such a way that the distances between vertices are equal to modified Hamming distances between corresponding strings.
A scheme for addressing digraphs is proposed, and the minimum address length is studied both in general and in certain special cases. The problem has some interesting reformulations in terms of matrix factorization and extremal set theory.