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.

What is the maximal number of edges in a bipartite graph of girth g whose left and right sides are of size nL , nR ? We generalize the known results for g = 6; 8 to an arbitrary girth. Keywords: graph, bipartite, girth, random walk. Institute of Computer Science, Hebrew University Jerusalem 91904 Israel shlomoh@cs.huji.ac.il 1 1

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...

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Abstract — Let C be an [n, k, d] binary linear code with rate R = k/n and dual C ⊥. In this work, it is shown that C can be represented by a 4-cycle-free Tanner graph only if: pd ⊥ ≤ $r np(p − 1) + n2

A two-dimensional grid is a set Gn,m = [n] × [m]. A grid Gn,m is c-colorable if there is a function χn,m: Gn,m → [c] such that there are no rectangles with all four corners the same color. We address the following question: for which values of n and m is Gn,m c-colorable? This problem can be viewed as a bipartite Ramsey problem and is related to a the Gallai-Witt theorem (also called the multidimensioanl Van Der Waerden’s Theorem). We determine (1) exactly which grids are 2-colorable, (2) exactly which grids are 3-colorable, and (3) (assuming a reasonable conjecture) exactly which grids are 4-colorable. We use combinatorics, finite fields, and tournament graphs.

In this paper, we study r-uniform hypergraphs without cycles of length less than five, employing the definition of a hypergraph cycle due to Berge. In particular, for r = 3, we show that if has n vertices and a maximum number of edges, then 1 ). This also asymptotically determines the generalized Turan number T 3 (n, 8, 4). Some results are based on our bounds for the maximum size of Sidon-type sets in n .

In the present paper @(n; 1‘1 will denote a graph of n vertices and 2 edges. K, will denote the complete graph of p vertices @ p; : and K(p, p) will denote the com-( 0) plete bipartite graph, more generally K(p,,..., pr) will denote the complete r-chromatic graph with pi vertices of the i-th colour, in which every two vertices of different colour are adjacent. C,, will denote a circuit having n edges. In 1940 T&N [l] posed and solved the following question: Determine the smallest integer m(n, p) so that every @(n; m(rz, p)) contains a Kp ’ Tudn in fact showed that the only @[n; m(n, p)- 1) which contains no K, is K(wt,,.... m,-z) where the mi are all as nearly equal as possible, i.e. for 0 5 i 5 p- 2 nti = [(n f i- l)j(p- l)]. Thus a simple computation gives that if n s r (mod p- 1) then +, P) = 2(“n-:, (n ’- r2) + (;) ’ Tursin further asked: How many edges must a graph contain that it should certainly have subgraphs of a prescribed structure? In particular he asked: Determine the smallest h(k, rt) so that every @(n; h(k, n)) should contain a path of length k. GALLAI and I [2] and ANDR~SFAI [2] investigated these and related questions and solved them nearly completely. In the present paper we shall try to investigate as systematically as possible the following question: What is the smallest integer f(n; k, Z) for which every graph B(n;f(n; k, I)) contains a @(k; 1) as a subgraph? These problems become very much more difficult, but in my belief also more interesting, if we also consider the structure of the graphs @(k; Z). W e now define three functions f i(n; k, I), 1 2 i 2 3.f,(n; k, 1) is the smallest integer for which every @(n;f,(n; k, I)) contains at least one @(k; I). f,(n; k, Z) is the smallest integer for which there is a 8(k; Z) of given structure so that every (Si(n;f,(rt; k, 1)) contains this @(k; I). f,(n; k, Z) is the smallest integer so that even the @(k; Z) which requires most edges occurs in

We construct an incidence structure using certain points and lines in finite projective spaces. The structural properties of the associated bipartite incidence graphs are analyzed. These n × n bipartite graphs provide constructions of C6-free graphs with Ω(n 4/3) edges, C10-free graphs with Ω(n 6/5) edges, and Θ(7, 7, 7)-free graphs with Ω(n 8/7) edges. Each of these bounds is sharp in order of magnitude.

Pooled Genomic Indexing (PGI) is a novel method for physical mapping of clones onto known sequences. PGI is