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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 ..."
Abstract

Cited by 205 (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
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.
The Chromatic Number of Graph Powers
, 2000
"... Introduction The square G 2 of a graph G = (V; E) is the graph whose vertex set is V in which two distinct vertices are adjacent if and only if their distance in G is at most 2. What is the maximum possible chromatic number of G 2 , as G ranges over all graphs with maximum degree d and girth g? ..."
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Cited by 15 (0 self)
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Introduction The square G 2 of a graph G = (V; E) is the graph whose vertex set is V in which two distinct vertices are adjacent if and only if their distance in G is at most 2. What is the maximum possible chromatic number of G 2 , as G ranges over all graphs with maximum degree d and girth g? Our (somewhat surprising) answer is that for g = 3; 4; 5 or 6 this maximum is (1 + o(1))d 2 (where the o(1) term tends to 0 as d tends to infinity), whereas for all g 7, this maximum is of order d 2 = log d. To state this result more precisely, define, for every two integers d 2 and g<F9.9
New constructions of bipartite graphs on m, n vertices with many edges and without small cycles
 Journal of Combinatorial Theory Ser. B
, 1994
"... For arbitrary odd prime power q and s ∈ (0, 1] such that q s is an integer, we construct a doubly–infinite series of (q 5, q 3+s)–bipartite graphs which are biregular of degrees q s and q 2 and of girth 8. These graphs have the greatest number of edges among all known (n, m)–bipartite graphs with t ..."
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Cited by 7 (4 self)
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For arbitrary odd prime power q and s ∈ (0, 1] such that q s is an integer, we construct a doubly–infinite series of (q 5, q 3+s)–bipartite graphs which are biregular of degrees q s and q 2 and of girth 8. These graphs have the greatest number of edges among all known (n, m)–bipartite graphs with the same asymptotics of log n m, n → ∞. For s = 1/3, our graphs provide an explicit counterexample to a conjecture of Erdős which states that an (n, m)–bipartite graph with m = O(n 2/3) and girth at least 8 has O(n) edges. This conjecture was recently disproved by de Caen and Székely [2], who established the existence of a family of such graphs having n 1+1/57+o(1) edges. Our graphs have n 1+1/15 edges, and so come closer to the best known upper bound of O(n 1+1/9). 1. Introduction. All graphs we consider are simple. The order (size) of a graph G is the number
Polarities and 2kcyclefree graphs
 Discrete Math
, 1999
"... Let C2k be the cycle on 2k vertices, and let ex(v, C2k) denote the greatest number of edges in a simple graph on v vertices which contains no subgraph isomorphic to C2k. In this paper we discuss a method which allows one to sometimes improve numerical constants in lower bounds for ex(v,C2k). The met ..."
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Cited by 6 (1 self)
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Let C2k be the cycle on 2k vertices, and let ex(v, C2k) denote the greatest number of edges in a simple graph on v vertices which contains no subgraph isomorphic to C2k. In this paper we discuss a method which allows one to sometimes improve numerical constants in lower bounds for ex(v,C2k). The method utilizes polarities in certain rank two geometries. It is applied to refute some conjectures about the values of ex(v,C2k), and to construct some new examples of graphs having certain restrictions on the lengths of their cycles. In particular, we construct an infinite family {Gi} of C6–free graphs with E(Gi)  ∼ 1 2 V (Gi)  4/3, i → ∞, which improves the constant in the previous best lower bound on ex(v,C6) from 2/3 4/3 ≈.462 to 1/2.
General properties of some families of graphs defined by systems of equations
 J. Graph Theory
, 2001
"... Abstract. In this paper we present a simple method for constructing infinite families of graphs defined by a class of systems of equations over commutative rings. We show that the graphs in all such families possess some general properties including regularity and biregularity, existence of special ..."
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Cited by 6 (4 self)
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Abstract. In this paper we present a simple method for constructing infinite families of graphs defined by a class of systems of equations over commutative rings. We show that the graphs in all such families possess some general properties including regularity and biregularity, existence of special vertex colorings, and existence of covering maps — hence, embedded spectra — between every two members of the same family. Another general property, recently discovered, is that nearly every graph constructed in this manner edgedecomposes either the complete, or complete bipartite, graph which it spans. In many instances, specializations of these constructions have proved useful in various graph theory problems, but especially in many extremal problems. A short survey of the related results is included. We also show that the edgedecomposition property allows one to improve existing lower bounds for some multicolor Ramsey numbers. 1.
An infinite series of regular edge but not vertextransitive graphs
 J. Graph Theory
, 2002
"... Abstract. Let n be an integer and q be a prime power. Then for any 3 ≤ n ≤ q − 1, or n = 2 and q odd, we construct a connected qregular edgebut not vertex transitive graph of order 2qn+1. This graph is defined via a system of equations over the finite field of q elements. For n = 2 and q = 3, our ..."
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Cited by 3 (2 self)
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Abstract. Let n be an integer and q be a prime power. Then for any 3 ≤ n ≤ q − 1, or n = 2 and q odd, we construct a connected qregular edgebut not vertex transitive graph of order 2qn+1. This graph is defined via a system of equations over the finite field of q elements. For n = 2 and q = 3, our graph is isomorphic to the Gray graph. 1.
On Extremal Graph Theory, Explicit Algebraic Constructions of Extremal Graphs and Corresponding Turing Encryption Machines
"... Abstract. We observe recent results on the applications of extremal graph theory to cryptography. Classical Extremal Graph Theory contains Erdős Even Circuite Theorem and other remarkable results on the maximal size of graphs without certain cycles. Finite automaton is roughly a directed graph with ..."
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Abstract. We observe recent results on the applications of extremal graph theory to cryptography. Classical Extremal Graph Theory contains Erdős Even Circuite Theorem and other remarkable results on the maximal size of graphs without certain cycles. Finite automaton is roughly a directed graph with labels on directed arrows. The most important advantage of Turing machine in comparison with finite automaton is existence of ”potentially infinite memory”. In terms of Finite Automata Theory Turing machine is an infinite sequence of directed graphs with colours on arrows. This is a motivation of studies of infinite families of extremal directed graphs without certain commutative diagrams. The explicite constructions of simple and directed graphs of large girth (or large cycle indicator) corresponds to efficient encryption of Turing machines.
On monomial graphs of girth eight
, 2005
"... Let e be a positive integer, p be an odd prime, q = pe,andFqbe the finite field of q elements. Let f2,f3 ∈ Fq[x,y]. The graph G = Gq(f2,f3) is a bipartite graph with vertex partitions P = F3 q and L = F3q,and edges defined as follows: a vertex (p) = (p1,p2,p3) ∈ P is adjacent to a vertex [l] =[l1, ..."
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Let e be a positive integer, p be an odd prime, q = pe,andFqbe the finite field of q elements. Let f2,f3 ∈ Fq[x,y]. The graph G = Gq(f2,f3) is a bipartite graph with vertex partitions P = F3 q and L = F3q,and edges defined as follows: a vertex (p) = (p1,p2,p3) ∈ P is adjacent to a vertex [l] =[l1,l2,l3] if and only if p2 + l2 = f2(p1,l1) and p3 + l3 = f3(p1,l1). Motivated by some questions in finite geometry and extremal graph theory, we ask when G has no cycle of length less than eight, i.e., has girth at least eight. When f2 and f3 are monomials, we call G a monomial graph. We show that for p � 5, and e = 2 a 3 b, a monomial graph of girth at least eight has to be isomorphic to the graph Gq(xy, xy 2), which is an induced subgraph of the classical generalized quadrangle W(q). For all other e, we show that a monomial graph is isomorphic to a graph Gq(xy, x k y 2k), with 1 � k � (q − 1)/2 and satisfying several other strong conditions. These conditions imply that k = 1forallq � 10 10. In particular, for a given positive integer k, the graph Gq(xy, x k y 2k) can be of girth eight only for finitely many odd characteristics p.