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34
Approximate distance oracles
 J. ACM
"... Let G = (V, E) be an undirected weighted graph with V  = n and E  = m. Let k ≥ 1 be an integer. We show that G = (V, E) can be preprocessed in O(kmn 1/k) expected time, constructing a data structure of size O(kn 1+1/k), such that any subsequent distance query can be answered, approximately, in ..."
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Cited by 207 (8 self)
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Let G = (V, E) be an undirected weighted graph with V  = n and E  = m. Let k ≥ 1 be an integer. We show that G = (V, E) can be preprocessed in O(kmn 1/k) expected time, constructing a data structure of size O(kn 1+1/k), such that any subsequent distance query can be answered, approximately, in O(k) time. The approximate distance returned is of stretch at most 2k − 1, i.e., the quotient obtained by dividing the estimated distance by the actual distance lies between 1 and 2k−1. A 1963 girth conjecture of Erdős, implies that Ω(n 1+1/k) space is needed in the worst case for any real stretch strictly smaller than 2k + 1. The space requirement of our algorithm is, therefore, essentially optimal. The most impressive feature of our data structure is its constant query time, hence the name “oracle”. Previously, data structures that used only O(n 1+1/k) space had a query time of Ω(n 1/k). Our algorithms are extremely simple and easy to implement efficiently. They also provide faster constructions of sparse spanners of weighted graphs, and improved tree covers and distance labelings of weighted or unweighted graphs. 1
Extensions of finite generalized quadrangles
 In Symposia Mathematica, Vol. XXVIII
, 1983
"... 1.1 Axioms and definitions.................................... 1 1.2 Restrictions on the parameters............................... 2 1.3 Regularity, antiregularity, semiregularity, and property (H)............... 3 ..."
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Cited by 109 (11 self)
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1.1 Axioms and definitions.................................... 1 1.2 Restrictions on the parameters............................... 2 1.3 Regularity, antiregularity, semiregularity, and property (H)............... 3
A new series of dense graphs of high girth
, 1995
"... Abstract. Let k ≥ 1 be an odd integer, t = ⌊ k+2 ⌋ , and q be a prime 4 power. We construct a bipartite, qregular, edgetransitive graph CD(k, q) of order v ≤ 2qk−t+1 and girth g ≥ k + 5. If e is the the number of edges 1+ 1 of CD(k, q) , then e = Ω(v k−t+1). These graphs provide the best known as ..."
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Cited by 40 (8 self)
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Abstract. Let k ≥ 1 be an odd integer, t = ⌊ k+2 ⌋ , and q be a prime 4 power. We construct a bipartite, qregular, edgetransitive graph CD(k, q) of order v ≤ 2qk−t+1 and girth g ≥ k + 5. If e is the the number of edges 1+ 1 of CD(k, q) , then e = Ω(v k−t+1). These graphs provide the best known asymptotic lower bound for the greatest number of edges in graphs of order v and girth at least g, g ≥ 5, g ̸ = 11, 12. For g ≥ 24, this represents a slight improvement on bounds established by Margulis and Lubotzky, Phillips, Sarnak; for 5 ≤ g ≤ 23, g ̸ = 11, 12, it improves on or ties existing bounds. 1.
Moore graphs and beyond: A survey of the degree/diameter problem
 ELECTRONIC JOURNAL OF COMBINATORICS
, 2013
"... The degree/diameter problem is to determine the largest graphs or digraphs of given maximum degree and given diameter. General upper bounds – called Moore bounds – for the order of such graphs and digraphs are attainable only for certain special graphs and digraphs. Finding better (tighter) upper bo ..."
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Cited by 26 (4 self)
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The degree/diameter problem is to determine the largest graphs or digraphs of given maximum degree and given diameter. General upper bounds – called Moore bounds – for the order of such graphs and digraphs are attainable only for certain special graphs and digraphs. Finding better (tighter) upper bounds for the maximum possible number of vertices, given the other two parameters, and thus attacking the degree/diameter problem ‘from above’, remains a largely unexplored area. Constructions producing large graphs and digraphs of given degree and diameter represent a way of attacking the degree/diameter problem ‘from below’. This survey aims to give an overview of the current stateoftheart of the degree/diameter problem. We focus mainly on the above two streams of research. However, we could not resist mentioning also results on various related problems. These include considering Moorelike bounds for special types of graphs and digraphs, such as vertextransitive, Cayley, planar, bipartite, and many others, on
Coverings, heat kernels and spanning trees
 ELECTRONIC JOURNAL OF COMBINATORICS
, 1999
"... We consider a graph G and a covering ˜ G of G and we study the relations of their eigenvalues and heat kernels. We evaluate the heat kernel for an infinite kregular tree and we examine the heat kernels for general kregular graphs. In particular, we show that a kregular graph on n vertices has at ..."
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Cited by 24 (10 self)
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We consider a graph G and a covering ˜ G of G and we study the relations of their eigenvalues and heat kernels. We evaluate the heat kernel for an infinite kregular tree and we examine the heat kernels for general kregular graphs. In particular, we show that a kregular graph on n vertices has at most (1 + o(1)) 2logn k−1
An Extremal Problem for Random Graphs and the Number of Graphs With Large EvenGirth
, 1995
"... We study the maximal number of edges a C2k free subgraph of a random graph Gn;p may have, obtaining best possible results for a range of p = p(n). Our estimates strengthen previous bounds of Furedi [12] and Haxell, Kohayakawa, and / Luczak [13]. Two main tools are used here: the first one is an ..."
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Cited by 22 (11 self)
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We study the maximal number of edges a C2k free subgraph of a random graph Gn;p may have, obtaining best possible results for a range of p = p(n). Our estimates strengthen previous bounds of Furedi [12] and Haxell, Kohayakawa, and / Luczak [13]. Two main tools are used here: the first one is an upper bound for the number of graphs with large evengirth, i.e., graphs without short even cycles, with a given number of vertices and edges, and satisfying a certain additional pseudorandom condition; the second tool is the powerful result of Ajtai, Koml'os, Pintz, Spencer, and Szemer'edi [1] on uncrowded hypergraphs as given by Duke, Lefmann, and Rodl [7].
On the locality of distributed sparse spanner construction
 In ACM Press, editor, 27th Annual ACM Symp. on Principles of Distributed Computing (PODC
, 2008
"... The paper presents a deterministic distributed algorithm that, given k � 1, constructs in k rounds a (2k−1, 0)spanner of O(kn 1+1/k)edgesforeverynnode unweighted graph. (If n is not available to the nodes, then our algorithm executes in 3k − 2 rounds, and still returns a (2k − 1, 0)spanner with O ..."
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Cited by 16 (7 self)
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The paper presents a deterministic distributed algorithm that, given k � 1, constructs in k rounds a (2k−1, 0)spanner of O(kn 1+1/k)edgesforeverynnode unweighted graph. (If n is not available to the nodes, then our algorithm executes in 3k − 2 rounds, and still returns a (2k − 1, 0)spanner with O(kn 1+1/k) edges.) Previous distributed solutions achieving such optimal stretchsize tradeoff either make use of randomization providing performance guarantees in expectation only, or perform in log Ω(1) n rounds, and all require a priori knowledge of n. Based on this algorithm, we propose a second deterministic distributed algorithm that, for every ɛ>0, constructs a (1 + ɛ, 2)spanner of O(ɛ −1 n 3/2)edgesin O(ɛ −1) rounds, without any prior knowledge on the graph. Our algorithms are complemented with lower bounds, which hold even under the assumption that n is known to the nodes. It is shown that any (randomized) distributed algorithm requires k rounds in expectation to compute a (2k − 1, 0)spanner of o(n 1+1/(k−1))edgesfork ∈{2, 3, 5}. It is also shown that for every k>1, any (randomized) distributed algorithm that constructs a spanner with fewer than n 1+1/k+ɛ edges in at most n ɛ expected rounds must stretch some distances by an additive factor of n Ω(ɛ).Inotherwords, while additive stretched spanners with O(n 1+1/k) edges may exist, e.g., for k =2, 3, they cannot be computed distributively in a subpolynomial number of rounds in expectation. Supported by the équipeprojet INRIA “DOLPHIN”. Supported by the ANRproject “ALADDIN”, and the
New Bounds on Crossing Numbers
, 1999
"... The crossing number, cr(G), of a graph G is the least number of crossing points in any drawing of G in the plane. Denote by #(n, e) the minimum of cr(G) taken over all graphs with n vertices and at least e edges. We prove a conjecture of P. Erdos and R. Guy by showing that #(n, e)n 2 /e 3 tends ..."
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Cited by 12 (4 self)
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The crossing number, cr(G), of a graph G is the least number of crossing points in any drawing of G in the plane. Denote by #(n, e) the minimum of cr(G) taken over all graphs with n vertices and at least e edges. We prove a conjecture of P. Erdos and R. Guy by showing that #(n, e)n 2 /e 3 tends to a positive constant as n ## and n # e # n 2 . Similar results hold for graph drawings on any other surface of fixed genus. We prove better bounds for graphs satisfying some monotone properties. In particular, we show that if G is a graph with n vertices and e # 4n edges, which does not contain a cycle of length four (resp. six), then its crossing number is at least ce 4 /n 3 (resp. ce 5 /n 4 ), where c > 0 is a suitable constant. These results cannot be improved, apart from the value of the constant. This settles a question of M. Simonovits. 1 Introduction Let G be a simple undirected graph with n(G) nodes (vertices) and e(G) edges. A drawing of G in the plane is a m...
Open problems of Paul Erdős in Graph Theory
 JOURNAL OF GRAPH THEORY
, 1997
"... The main treasure that Paul Erdős has left us is his collection of problems, most of which are still open today. These problems are seeds that Paul sowed and watered by giving numerous talks at meetings big and small, near and far. In the past, his problems have spawned many areas in graph theory an ..."
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Cited by 11 (0 self)
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The main treasure that Paul Erdős has left us is his collection of problems, most of which are still open today. These problems are seeds that Paul sowed and watered by giving numerous talks at meetings big and small, near and far. In the past, his problems have spawned many areas in graph theory and beyond (e.g., in number theory, probability, geometry, algorithms and complexity theory). Solutions or partial solutions to Erdős problems usually lead to further questions, often in new directions. These problems provide inspiration and serve as a common focus for all graph theorists. Through the problems, the legacy of Paul Erdős continues (particularly if solving one of these problems results in creating three new problems, for example.) There is a huge literature of almost 1500 papers written by Erdős and his (more than 460) collaborators. Paul wrote many problem papers, some of which appeared in various (really hardtofind) proceedings. Here is an attempt to collect and organize these problems in the area of graph theory. The list here is by no means complete or exhaustive. Our goal is to state the problems, locate the sources, and provide the references related to these problems. We will include the earliest and latest known references without covering the entire history of the problems because of space limitations. (The most uptodate list of Erdős’ papers can be found in [65]; an electronic file is maintained by Jerry Grossman at
Large Cayley Graphs and Digraphs with Small Degree and Diameter
, 1995
"... We review the status of the Degree#Diameter problem for both, graphs and digraphs and present new Cayley digraphs which yield improvements over some of the previously known largest vertex transitive digraphs of given degree and diameter. ..."
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Cited by 10 (0 self)
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We review the status of the Degree#Diameter problem for both, graphs and digraphs and present new Cayley digraphs which yield improvements over some of the previously known largest vertex transitive digraphs of given degree and diameter.