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129
Books in graphs
, 2008
"... 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 wi ..."
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Cited by 1802 (19 self)
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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
Wireless Network Information Flow: A Deterministic Approach
, 2009
"... In contrast to wireline networks, not much is known about the flow of information over wireless networks. The main barrier is the complexity of the signal interaction in wireless channels in addition to the noise in the channel. A widely accepted model is the the additive Gaussian channel model, and ..."
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Cited by 88 (15 self)
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In contrast to wireline networks, not much is known about the flow of information over wireless networks. The main barrier is the complexity of the signal interaction in wireless channels in addition to the noise in the channel. A widely accepted model is the the additive Gaussian channel model, and for this model, the capacity of even a network with a single relay node is open for 30 years. In this paper, we present a deterministic approach to this problem by focusing on the signal interaction rather than the noise. To this end, we propose a deterministic channel model which is analytically simpler than the Gaussian model but still captures two key wireless channel properties of broadcast and superposition. We consider a model for a wireless relay network with nodes connected by such deterministic channels, and present an exact characterization of the endtoend capacity when there is a single source and one or more destinations (all interested in the same information) and an arbitrary number of relay nodes. This result is a natural generalization of the celebrated maxflow mincut theorem for wireline networks. We then use the insights obtained from the analysis of the deterministic model to study information flow for the Gaussian wireless relay network. We present an achievable rate for general Gaussian relay networks and show that it is within a constant number of bits from the cutset bound on the capacity of these networks. This constant depends on the number of nodes in the network, but not the values of the channel gains or the signaltonoise ratios. We show that existing strategies cannot achieve such a constantgap approximation for arbitrary networks and propose a new quantizemapandforward scheme that does. We also give several extensions of the approximation framework including robustness results (through compound channels), halfduplex constraints and ergodic channel variations.
Towards a theory of scalefree graphs: Definition, properties, and implications
 Internet Mathematics
, 2005
"... Abstract. There is a large, popular, and growing literature on “scalefree ” 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 ..."
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Cited by 83 (8 self)
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Abstract. There is a large, popular, and growing literature on “scalefree ” 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 scalefree 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 scalefree. This structural view can be related to previously studied graph properties such as the various notions of selfsimilarity, 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 theoremproof style of exposition, but who may be unfamiliar with the existing literature on scalefree networks. 1.
Proving Integrality Gaps Without Knowing the Linear Program
 Theory of Computing
, 2002
"... Proving integrality gaps for linear relaxations of NP optimization problems is a difficult task and usually undertaken on a casebycase basis. We initiate a more systematic approach. We prove an integrality gap of 2o(1) for three families of linear relaxations for vertex cover, and our methods see ..."
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Cited by 56 (2 self)
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Proving integrality gaps for linear relaxations of NP optimization problems is a difficult task and usually undertaken on a casebycase basis. We initiate a more systematic approach. We prove an integrality gap of 2o(1) for three families of linear relaxations for vertex cover, and our methods seem relevant to other problems as well.
Girth of Sparse Graphs
 2002), 194  200. ILWOO CHO
"... 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 nonnegative reals, although the d ..."
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Cited by 49 (4 self)
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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 nonnegative reals, although the details are much more complicated, to ensure the exact inclusion of many of the recent models for largescale realworld 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 quasirandom graphs are in a
The Interlace Polynomial of a Graph
, 2004
"... Motivated by circle graphs, and the enumeration of Euler circuits, we define a onevariable “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 t ..."
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Cited by 39 (1 self)
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Motivated by circle graphs, and the enumeration of Euler circuits, we define a onevariable “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 twovariable 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.
Frozen Development in Graph Coloring
 THEORETICAL COMPUTER SCIENCE
, 2000
"... We define 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 first describe in detail the algori ..."
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Cited by 34 (5 self)
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We define 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 first describe in detail the algorithmic techniques used to study frozen development. We present strong empirical evidence that freezing in 3coloring 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...
The Interlace Polynomial: A New Graph Polynomial
 JOURNAL OF COMBINATORIAL THEORY, SERIES B
, 2000
"... 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 welldefined; in the absence of a direct interpretation of the polynomial, this remains mysterious. The interlace pol ..."
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Cited by 31 (3 self)
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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 welldefined; 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 2in, 2out 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...
The Kauffman Bracket of Virtual Links and the BolloásRiordan Polynomial
 the Moscow Mathematical Journal. Preprint arXiv:math.GT/0609012
"... Abstract. We show that the Kauffman bracket [L] of a checkerboard colorable virtual link L is an evaluation of the Bollobás–Riordan polynomial RG L of a ribbon graph associated with L. This result generalizes the celebrated relation between the classical Kauffman bracket and the Tutte polynomial of ..."
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Cited by 25 (2 self)
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Abstract. We show that the Kauffman bracket [L] of a checkerboard colorable virtual link L is an evaluation of the Bollobás–Riordan polynomial RG L of a ribbon graph associated with L. This result generalizes the celebrated relation between the classical Kauffman bracket and the Tutte polynomial of planar graphs.