The Deletion Method For Upper Tail Estimates

We present a new method to show concentration of the upper tail of random variables that can be written as sums of variables with plenty of independence. We compare our method with the martingale method by Kim and Vu, which often leads to similar results.Some applications are given to the number X _G of copies of a graph G in the random graph (n,p). In particular, for G = K ₄ and G = C ₄ we improve the earlier known upper bounds on —ln (X _G 2 X _G ) in some range of p = p(n).* Research of the second author supported by KBN grants 2 P03A 032 16 and 2 P03A 015 23.     

Isoperimetric numbers of graph bundles

The main aim of this paper is to give some upper and lower bounds for the isoperimetric numbers of graph coverings or graph bundles, with exact values in some special cases. In addition, we show that the isoperimetric number of any covering graph is not greater than that of the base graph. Mohar's theorem for the isoperimetric number of the cartesian product of a graph and a complete graph can be extended to a more general case: The isoperimetric numberi(G × K _2n) of the cartesian product of any graphG and a complete graphK _2n on even vertices is the minimum of the isoperimetric numberi(G) andn, and it is also a sharp lower bound of the isoperimetric numbers of all graph bundles over the graphG with fiberK _2n. Furthermore, ifn 2i(G) then the isoperimetric number of any graph bundle overG with fibreK _n is equal to the isoperimetric numberi(G) ofG. Partially supported by The Ministry of Education, Korea.     

Large induced degenerate subgraphs

A graphH isd-degenerate if every subgraph of it contains a vertex of degree smaller thand. For a graphG, let _d (G) denote the maximum number of vertices of an inducedd-degenerate subgraph ofG. Sharp lowers bounds for _d (G) in terms of the degree sequence ofG are obtained, and the minimum number of edges of a graphG withn vertices and ₂ (G) m is determined precisely for allm n. Research supported in part by an Allon Fellowship and by a Bat-Sheva de Rothschild grant.Research supported in part by NSF grant MCS 8301867, and by a Sloan Research Fellowship.     

Vaught’s conjecture for some meager groups

AssumeG is a superstable group ofM-rank 1 and the division ring of pseudo-endomorphisms ofG is a prime field. We prove a relative Vaught’s conjecture for Th(G). When additionallyU(G) =ω, this yields Vaught’s conjecture for Th(G). Research supported by KBN grant 2 P03A 006 09.     

Edge-clique graphs

The edge-clique graphK(G) of a graphG is that graph whose vertices correspond to the edges ofG and where two vertices ofK(G) are adjacent whenever the corresponding edges ofG belong to a common clique. It is shown that every edge-clique graph is a clique graph, and that ifG is either an interval graph or a line graph, then so too isK(G). An algorithm is provided for determining whether a graph is an edge-clique graph. A new graph called the STP graph is introduced and a relationship involving this graph, the edge-clique graph, and the line graph is presented. The STP graphs are also characterized.Research supported in part by Office of Naval Research Contract N00014-88-K-0018.Research supported in part by Office of Naval Research Contract N00014-88-K-0163.     

Eigenvalues,diameter, and mean distance in graphs

It is well-known that the second smallest eigenvalue ₂ of the difference Laplacian matrix of a graphG is related to the expansion properties ofG. A more detailed analysis of this relation is given. Upper and lower bounds on the diameter and the mean distance inG in terms of ₂ are derived.This work was supported in part by the Research Council of Slovenia, Yugoslavia. A part of the work was done while the author was visiting the Ohio State University, supported by a Fulbright grant.     

Code length between Markov processes

LetX ₁ andX ₂ be two mixing Markov shifts over finite alphabet. If the entropy ofX ₁ is strictly larger than the entropy ofX ₂, then there exists a finitary homomorphism ϕ:X ₁→X ₂ such that the code length is anL ^p random variable for allp<4/3. In particular, the expected length of the code ϕ is finite. Research supported by KBN grant 2 P03A 039 15 1998–2001.     

Sum-free sets in abelian groups

We show that there is an absolute constant δ>0 such that the number of sum-free subsets of any finite abelian groupG is

Opposition graphs are strict quasi-parity graphs

An opposition graph is a graph whose edges can be acyclically oriented in such a way that every chordless path on four vertices has its extreme edges both pointing in or pointing out. A strict quasi-parity graph is a graphG such that every induced subgraphH ofG either is a clique or else contains a pair of vertices which are not endpoints of an odd (number of edges) chordless path ofH. The perfection of opposition graphs and strict quasi-parity graphs was established respectively by Olariu and Meyniel. We show here that opposition graphs are strict quasi-parity graphs.The second author acknowledges the support of the Air Force Office of Scientific Research under grant number AFOSR 0271 to Rutgers University.     

Facets of the balanced (acyclic) induced subgraph polytope

A signed graph is a graph whose edges are labelled positive or negative. A signed graph is said to be balanced if the set of negative edges form a cut. The balanced induced subgraph polytopeP(G) of a graphG is the convex hull of the incidence vectors of all node sets that induce balanced subgraphs ofG. In this paper we exhibit various (rank) facet defining inequalities. We describe several methods with which new facet defining inequalities ofP(G) can be constructed from known ones. Finding a maximum weighted balanced induced subgraph of a series parallel graph is a polynomial problem. We show that for this class of graphsP(G) may have complicated facet defining inequalities. We derive analogous results for the polytope of acyclic induced subgraphs.Research supported in part by the Natural Sciences and Engineering Research Council of Canada; the second author has also been supported by C.P. Rail.     

Relaxed chromatic numbers of graphs

Given any family of graphsP, theP chromatic number _p (G) of a graphG is the smallest number of classes into whichV(G) can be partitioned such that each class induces a subgraph inP. We study this for hereditary familiesP of two broad types: the graphs containing no subgraph of a fixed graphH, and the graphs that are disjoint unions of subgraphs ofH. We generalize results on ordinary chromatic number and we computeP chromatic number for special choices ofP on special classes of graphs.Research supported in part by ONR Grant N00014-85K0570 and by a grant from the University of Illinois Research Board.     

Perfect couples of graphs

We generalize the concept of perfect graphs in terms of additivity of a functional called graph entropy. The latter is an information theoretic functional on a graphG with a probability distributionP on its vertex set. For any fixedP it is sub-additive with respect to graph union. The entropy of the complete graph equals the sum of those ofG and its complement G iffG is perfect. We generalize this recent result to characterize all the cases when the sub-additivity of graph entropy holds with equality.The research of the authors is partially supported by the Hungarian National Foundation for Scientific Research (OTKA), grant No. 1806 resp. No. 1812.     

Spanning trees and a conjecture of Kontsevich

Kontsevich conjectured that the number of zeros over the fieldF _q of a certain polynomialQ _G associated with the spanning trees of a graphG is a polynomial function ofq. We show the connection between this conjecture, the Matrix-Tree Theorem, and orthogonal geometry. We verify the conjecture in certain cases, such as the complete graph, and discuss some modifications and extensions.Partially supported by NSF grant #DMS-9743966.     

Three-regular path pairable graphs

A graphG with at least 2k vertices isk-path pairable if for anyk pairs of distinct vertices ofG there arek edge disjoint paths between the pairs. It will be shown for any positive integerk that there is ak-path pairable graph of maximum degree three.Research is partially supported by ONR research grant N000014-88-K-0070 and NAS Exchange grant.     

Embedding finite graphs into graphs colored with infinitely many colors

We prove that for any cardinalτ and for any finite graphH there is a graphG such that for any coloring of the pairs of vertices ofG withτ colors there is always a copy ofH which is an induced subgraph ofG so that both the edges of the copy and the edges of the complement of the copy are monochromatic. Research supported by Hungarian National Science Foundation OTKA grant 1805.     

Greatest common divisors and least common multiples of graphs

A graphH divides a graphG, writtenH|G, ifG isH-decomposable. A graphG without isolated vertices is a greatest common divisor of two graphsG ₁ andG ₂ ifG is a graph of maximum size for whichG|G ₁ andG|G ₂, while a graphH without isolated vertices is a least common multiple ofG ₁ andG ₂ ifH is a graph of minimum size for whichG ₁|H andG ₂|H. It is shown that every two nonempty graphs have a greatest common divisor and least common multiple. It is also shown that the ratio of the product of the sizes of a greatest common divisor and least common multiple ofG ₁ andG ₂ to the product of their sizes can be arbitrarily large or arbitrarily small. Sizes of least common multiples of various pairsG ₁,G ₂ of graphs are determined, including when one ofG ₁ andG ₂ is a cycle of even length and the other is a star.G. C's research was supported in part by the Office of Naval Research, under Grant N00014-91-I-1060     

A correlation inequality for bipartite graphs

We conjecture an integral inequality for a product of functionsh(x _i ,y _j ) where the diagram of the product is a bipartite graphG. In particular, this inequality states that the random graph with fixed numbers of vertices and edges contains the asymptotically minimal number of copies ofG.     

H-extension of graphs

We consider the problem of constructing a graphG* from a collection of isomorphic copies of a graphG in such a way that for every two copies ofG, either no vertices or a section graph isomorphic to a graphH is identified. It is shown that ifG can be partitioned into vertex-disjoint copies ofH, thenG* can be made to have at most |H| orbits. A condition onG so thatG* can be vertextransitive is also included.     

Can visibility graphs Be represented compactly?

We consider the problem of representing the visibility graph of line segments as a union of cliques and bipartite cliques. Given a graphG, a familyG={G ₁,G ₂,...,G _k } is called aclique cover ofG if (i) eachG _i is a clique or a bipartite clique, and (ii) the union ofG _i isG. The size of the clique coverG is defined as ∑ _i=1 ^k n _i , wheren _i is the number of vertices inG _i . Our main result is that there are visibility graphs ofn nonintersecting line segments in the plane whose smallest clique cover has size Ω(n ²/log² n). An upper bound ofO(n ²/logn) on the clique cover follows from a well-known result in extremal graph theory. On the other hand, we show that the visibility graph of a simple polygon always admits a clique cover of sizeO(nlog³ n), and that there are simple polygons whose visibility graphs require a clique cover of size Ω(n logn). The work by the first author was supported by National Science Foundation Grant CCR-91-06514. The work by the second author was supported by a USA-Israeli BSF grant. The work by the third author was supported by National Science Foundation Grant CCR-92-11541.