Let G = (V; E) be an undirected weighted graph with jV j = n and jEj = m. Let k 1 be an integer. We show that G = (V; E) can be preprocessed in O(kmn ) expected time, constructing a data structure of size O(kn ), such that any subsequent distance query can be answered, approximately, in O(k) time. The approximate distance returned is of stretch at most 2k \Gamma 1, i.e., the quotient obtained by dividing the estimated distance by the actual distance lies between 1 and 2k \Gamma 1. We show that a 1963 girth conjecture of Erdos, implies ) 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.

Abstract. Let k ≥ 1 be an odd integer, t = ⌊ k+2 ⌋ , and q be a prime 4 power. We construct a bipartite, q-regular, edge-transitive 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.

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

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 bireg-ular 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

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.

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 bi-regularity, 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 edge-decomposes 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.

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