An extremal problem for Graham-Rothschild parameter words

This paper exposes connections between the theory of Möbius functions and extremal problems, extending ideas of Frankl and Pach [8]. Extremal results concerning the trace of objects in geometric lattices and Graham—Rothschild parameter posets are proved, covering previous results due to Sauer [16] and Perles and Shelah [17].     

Boundedness of optimal matrices in extremal multigraph and digraph problems

We consider multigraphs in which any two vertices are joined by at mostq edges, and study the Turán-type problem for a given family of forbidden multigraphs. In the caseq=2, answering a question of Brown, Erds 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, Erds and Simonovits.     

The splittance of a graph

The splittance of an arbitrary graph is the minimum number of edges to be added or removed in order to produce a split graph (i.e. a graph whose vertex set can be partitioned into a clique and an independent set). The splittance is seen to depend only on the degree sequence of the graph, and an explicit formula for it is derived. This result allows to give a simple characterization of the degree sequences of split graphs. Worst cases for the splittance are determined for some classes of graphs (the class of all graphs, of all trees and of all planar graphs).     

A strongly regularN-full graph of small order

For every positive integern we show the construction of a strongly regular graph of order at most 2 ⁿ⁺² which contains every graph of ordern as a subgraph. The estimation concerning the construction is best possible.     

On the number ofC 5′S in a triangle-free graph

Research partially supported bv Hungarian National Foundation for Scientific Research Grant no. 1812.     

Bounding the number of circuits of a graph

Letc(G) denote the number of circuits of a graphG. In this paper, we characterize those minor-closed classesG of graphs for which there is a polynomial functionp(.) such thatc(G)p(|E(G)|) for all graphsG inG.     

Infinite spectra in the first order theory of graphs

The psectrum Spec(A) of a sentenceA is, roughly, the set of those a for whichA has a threshold function at or nearp=n ^–a . Examples are given ofA with infinite spectra and with spectra of order type ⁱ for arbitraryi.     

Further results on monotonic graph invariants and bipartiteness number

Abstract

The bipartiteness of a graph is the minimum number of vertices whose deletion from G results in a bipartite graph. If a graph invariant decreases or increases with addition of edges of its complement, then it is called a monotonic graph invariant. In this article, we determine the extremal values of some famous monotonic graph invariants, and characterize the corresponding extremal graphs in the class of all connected graphs with a given vertex bipartiteness.     

On the Szeged and the Laplacian Szeged spectrum of a graph

For a given graph G its Szeged weighting is defined by w(e)=n_u(e)n_v(e), where e=uv is an edge of G,n_u(e) is the number of vertices of G closer to u than to v, and n_v(e) is defined analogously. The adjacency matrix of a graph weighted in this way is called its Szeged matrix. In this paper we determine the spectra of Szeged matrices and their Laplacians for several families of graphs. We also present sharp upper and lower bounds on the eigenvalues of Szeged matrices of graphs.     

Disproof of a conjecture in graph reconstruction theory

In his thesis [3] B. D. Thatte conjectured that ifG=G ₁,G ₂,...G _n is a sequence of finitely many simple connected graphs (isomorphic graphs may occur in the sequence) with the same number of vertices and edges then their shuffled edge deck uniquely determines the graph sequence (up to a permutation). In this paper we prove that there are such sequences of graphs with the same shuffled edge deck.This research was partially supported by Hungarian National Foundation of Scientific Research Grant no. 1812     

Low diameter graph decompositions

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 . 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(log² n). 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.A preliminary version of this paper appeared as Decomposing Graphs into Regions of Small Diameter in Proc. 2nd ACM-SIAM Symposium on Discrete Algorithms (1991) 321-330.This work was supported in part by NSF grant DMS87-03541 and by a grant from the Israel Academy of Science.This work was supported in part by NSF grant DMS87-03541 and CCR89-11388.     

On the topology of graph picture spaces

We study the space of pictures of a graph G in complex projective d-space. The main result is that the homology groups (with integer coefficients) of are completely determined by the Tutte polynomial of G. One application is a criterion in terms of the Tutte polynomial for independence in the d-parallel matroids studied in combinatorial rigidity theory. For certain special graphs called orchards, the picture space is smooth and has the structure of an iterated projective bundle. We give a Borel presentation of the cohomology ring of the picture space of an orchard, and use this presentation to develop an analogue of the classical Schubert calculus.     

How to draw a planar graph on a grid

Answering a question of Rosenstiehl and Tarjan, we show that every plane graph withn vertices has a Fáry embedding (i.e., straight-line embedding) on the 2n–4 byn–2 grid and provide anO(n) space,O(n logn) time algorithm to effect this embedding. The grid size is asymptotically optimal and it had been previously unknown whether one can always find a polynomial sized grid to support such an embedding. On the other hand we show that any setF, which can support a Fáry embedding of every planar graph of sizen, has cardinality at leastn+(1–o(1))n which settles a problem of Mohar.Supported in part by P. R. C. Mathematiques et Informatique.Supported in part by HSF grant 1814.Part of the work on this paper was done while this author was on sabbatical leave at école Normal Supérieure and école des Hautes études en Sciences Sociales; supported in part by NSF grant DMS-850 1947.     

On a random graph evolving by degrees

We consider a random (multi)graph growth process {Gm} on a vertex set [n], which is a special case of a more general process proposed by Laci Lovász in 2002. G₀ is empty, and Gm+1 is obtained from Gm by inserting a new edge e at random. Specifically, the conditional probability that e joins two currently disjoint vertices, i and j, is proportional to (di+α)(dj+α), where di, dj are the degrees of i, j in Gm, and α>0 is a fixed parameter. The limiting case α=∞ is the Erd?s-Rényi graph process. We show that whp Gm contains a unique giant component iff c:=2m/n>cα=α/(1+α), and the size of this giant is asymptotic to , where c^∗<cα is the root of . A phase transition window is proved to be contained, essentially, in [cα−An−1/3,cα+Bn−1/4], and we conjecture that 1/4 may be replaced with 1/3. For the multigraph version, {MGm}, we show that MGm is connected whp iff m?mn:=n1+α−1. We conjecture that, for α>1, mn is the threshold for connectedness of Gm itself.     

Applications of a theorem by Ky Fan in the theory of graph energy

The energy of a graph G is equal to the sum of the absolute values of the eigenvalues of G, which in turn is equal to the sum of the singular values of the adjacency matrix of G. Let X, Y, and Z be matrices, such that X+Y=Z. The Ky Fan theorem establishes an inequality between the sum of the singular values of Z and the sum of the sum of the singular values of X and Y. This theorem is applied in the theory of graph energy, resulting in several new inequalities, as well as new proofs of some earlier known inequalities.     

How to make a graph four-connected

(i) every 2-connected graph on n vertices can be made 4-connected by adding at most n new edges, and (ii) every 3-connected and 3-regular graph on n≥8 vertices can be made 4-connected by adding n/2 new edges. Received October 1995 / Revised version received March 1997 Published online March 16, 1999     

Thep-intersection number of a complete bipartite graph and orthogonal double coverings of a clique

Thep-intersection graph of a collection of finite sets {S _i } _i=1 ⁿ is the graph with vertices 1, ...,n such thati, j are adjacent if and only if |S _i S _j |p. Thep-intersection number of a graphG, herein denoted _p (G), is the minimum size of a setU such thatG is thep-intersection graph of subsets ofU. IfG is the complete bipartite graphK _n,n andp2, then _p (K _{n, n} )(n ²+(2p–1)n)/p. Whenp=2, equality holds if and only ifK _n has anorthogonal double covering, which is a collection ofn subgraphs ofK _n , each withn–1 edges and maximum degree 2, such that each pair of subgraphs shares exactly one edge. By construction,K _n has a simple explicit orthogonal double covering whenn is congruent modulo 12 to one of {1, 2, 5, 7, 10, 11}.Research supported in part by ONR Grant N00014-5K0570.     

A geometric theory of hypergraph colouring

Summary Ak-colouring of a hypergraph is an assignment of no more thank colours to the vertices of the hypergraph in such a way that the coloured hypergraph has no monochromatic edges. In this paper a polymatroid is associated with a hypergraph. It is shown that the number ofk-colourings of the hypergraph is determined by an evaluation of the characteristic polynomial of this polymatroid. Hypergraph colouring is then related to an extension of the critical problem developed by the author. It is shown that, ifq is a prime power, then a hypergraph isq ^k -colourable if and only if the critical exponent overGF(q) of its associated polymatroid is less than or equal tok.