Results 1  10
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18
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 279 (10 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 50 (10 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.
General properties of some families of graphs defined by systems of equations
, 2003
"... 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 ..."
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Cited by 10 (5 self)
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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.
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 9 (6 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.
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 8 (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.
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
Properties of certain families of 2kcycle free graphs
 J. COMBIN. THEORY, SER. B
, 1994
"... Let v = v(G) and e = e(G) denote the order and size of a simple graph G, respectively. Let G = {Gi}i≥1 be a family of simple graphs of magnitude r> 1 and constant λ> 0, i.e. e(Gi) = (λ + o(1))v(Gi) r, i → ∞. For any such family G whose members are bipartite and of girth at least 2k + 2, and e ..."
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Cited by 6 (4 self)
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Let v = v(G) and e = e(G) denote the order and size of a simple graph G, respectively. Let G = {Gi}i≥1 be a family of simple graphs of magnitude r> 1 and constant λ> 0, i.e. e(Gi) = (λ + o(1))v(Gi) r, i → ∞. For any such family G whose members are bipartite and of girth at least 2k + 2, and every integer t, 2 ≤ t ≤ k − 1, we construct a family ˜ Gt of graphs of same magnitude r, of constant greater than λ, and all of whose members contain each of the cycles C4, C6,..., C2t, but none of the cycles C2t+2,..., C2k. We also prove that for every family of 2k–cycle free extremal graphs (i.e. graphs having the greatest size among all 2k–cycle free graphs of the same order), all but finitely many such graphs must be either non–bipartite or have girth at most 2k − 2. In particular, we show that the best known lower bound on the size of 2k–cycle k+1 free extremal graphs for k = 3, 5, namely (2 k + o(1))v k+1 k, can be improved k+1 to ((k − 1) · k k + o(1))v k+1 k.
Romańczuk U.: On Extremal Graph Theory, Explicit Algebraic Constructions of Extremal Graphs and Corresponding Turing Encryption
 Machines, Artificial Intelligence, Evolutionary Computing and Metaheuristics, In the footsteps of Alan Turing Series: Studies in Computational Intelligence
, 2013
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