A set of q triangles sharing a common edge is called a book of size q. We write β (n, m) for the the maximal q such that every graph G (n, m) contains a book of size q. In this note 1) we compute β ( n, cn 2) for infinitely many values of c with 1/4 < c < 1/3, 2) we show that if m ≥ (1/4 − α) n 2 with 0 < α < 17 −3 (), and G has no book of size at least graph G1 of order at least

Proving integrality gaps for linear relaxations of NP optimization problems is a difficult task and usually undertaken on a case-by-case basis. We initiate a more systematic approach. We prove an integrality gap of 2-o(1) for three families of linear relaxations for vertex cover, and our methods seem relevant to other problems as well.

Abstract. There is a large, popular, and growing literature on “scale-free ” networks with the Internet along with metabolic networks representing perhaps the canonical examples. While this has in many ways reinvigorated graph theory, there is unfortunately no consistent, precise definition of scale-free graphs and few rigorous proofs of many of their claimed properties. In fact, it is easily shown that the existing theory has many inherent contradictions and that the most celebrated claims regarding the Internet and biology are verifiably false. In this paper, we introduce a structural metric that allows us to differentiate between all simple, connected graphs having an identical degree sequence, which is of particular interest when that sequence satisfies a power law relationship. We demonstrate that the proposed structural metric yields considerable insight into the claimed properties of SF graphs and provides one possible measure of the extent to which a graph is scale-free. This structural view can be related to previously studied graph properties such as the various notions of self-similarity, likelihood, betweenness and assortativity. Our approach clarifies much of the confusion surrounding the sensational qualitative claims in the current literature, and offers a rigorous and quantitative alternative, while suggesting the potential for a rich and interesting theory. This paper is aimed at readers familiar with the basics of Internet technology and comfortable with a theorem-proof style of exposition, but who may be unfamiliar with the existing literature on scale-free networks. 1.

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Recently, Bollobás, Janson and Riordan introduced a very general family of random graph models, producing inhomogeneous random graphs with Θ(n) edges. Roughly speaking, there is one model for each kernel, i.e., each symmetric measurable function from [0,1] 2 to the non-negative reals, although the details are much more complicated, to ensure the exact inclusion of many of the recent models for large-scale real-world networks. A different connection between kernels and random graphs arises in the recent work of Borgs, Chayes, Lovász, Sós, Szegedy and Vesztergombi. They introduced several natural metrics on dense graphs (graphs with n vertices and Θ(n 2) edges), showed that these metrics are equivalent, and gave a description of the completion of the space of all graphs with respect to any of these metrics in terms of graphons, which are essentially bounded kernels. One of the most appealing aspects of this work is the message that sequences of inhomogeneous quasi-random graphs are in a

We dene the `frozen development' of coloring random graphs. We identify two nodes in a graph as frozen if they are the same color in all legal colorings. This is analogous to studies of the development of a backbone or spine in SAT (the Satisability problem). We rst describe in detail the algorithmic techniques used to study frozen development. We present strong empirical evidence that freezing in 3-coloring is sudden. A single edge typically causes the size of the graph to collapse in size by 28%. We also use the frozen development to calculate unbiased estimates of probability of colorability in random graphs, even where this probability is as low as 10 300 . We investigate the links between frozen development and the solution cost of graph coloring. In SAT, a discontinuity in the order parameter has been correlated with the hardness of SAT instances, and our data for coloring is suggestive of an asymptotic discontinuity. The uncolorability threshold is known to give rise to har...

Motivated by circle graphs, and the enumeration of Euler circuits, we define a one-variable “interlace polynomial ” for any graph. The polynomial satisfies a beautiful and unexpected reduction relation, quite different from the cut and fuse reduction characterizing the Tutte polynomial. It emerges that the interlace graph polynomial may be viewed as a special case of the Martin polynomial of an isotropic system, which underlies its connections with the circuit partition polynomial and the Kauffman brackets of a link diagram. The graph polynomial, in addition to being perhaps more broadly accessible than the Martin polynomial for isotropic systems, also has a two-variable generalization that is unknown for the Martin polynomial. We consider extremal properties of the interlace polynomial, its values for various special graphs, and evaluations which relate to basic graph properties such as the component and independence numbers.

We define a new graph polynomial, the interlace polynomial, for any undirected graph. One of our main results is that the polynomial, specified by an intricate recursion relation, is well-defined; in the absence of a direct interpretation of the polynomial, this remains mysterious. The interlace polynomial is not a special case of the Tutte polynomial, nor do we know of any other graph polynomial to which it can be reduced. For 2-in, 2-out directed graphs D, any Euler circuit induces an undirected "interlace" graph H. In this setting, the interlace polynomial q(H) is equal to the Martin polynomial m(D), a variant of the circuit partition polynomial. There is another connection, between the Kauffman brackets of a link diagram, the Martin polynomial, and in turn the interlace polynomial. We explore other properties of the interlace polynomial, such as its relations with the component number and independence number of a graph, its extremal values, and its values for various classes of g...

Analyzing the worst-case complexity of the k-level in a planar arrangement of n curves is a fundamental problem in combinatorial geometry. We give the first subquadratic upper bound (roughly O(nk 9 2 s 3 )) for curves that are graphs of polynomial functions of an arbitrary fixed degree s. Previously, nontrivial results were known only for the case s = 1 and s = 2. We also improve the earlier bound for pseudo-parabolas (curves that pairwise intersect at most twice) to O(nk k). The proofs are simple and rely on a theorem of Tamaki and Tokuyama on cutting pseudo-parabolas into pseudo-segments, as well as a new observation for cutting pseudo-segments into pieces that can be extended to pseudo-lines. We mention applications to parametric and kinetic minimum spanning trees.

Abstract. We show that if X is the complement of a complex hyperplane arrangement, then the homology of X has linear free resolution as a module over the exterior algebra on the first cohomology of X. We study invariants of X that can be deduced from this resolution. A key ingredient is a result of Aramova, Avramov, and Herzog (2000) on resolutions of monomial ideals in the exterior algebra. We give a new conceptual proof of this result. Let X be the complement of a complex hyperplane arrangement A. Inthispaper we study the singular homology H∗(X) as a module over the exterior algebra E on the first singular cohomology V: = H1 (X) always with coefficients in a fixed field K. Our first main result (Section 1) asserts that H∗(X) is generated in a single degree and has a linear free resolution; this amounts to an infinite sequence of statements about the multiplication in the Orlik-Solomon algebra H ∗ (X). We also analyze other topological examples from the point of view of resolutions over the exterior algebra. In Section 2 we study an invariant of an E-module N called the singular variety,