Extremal graphs with no C4's, C6's, or C10's

Wenger, Rephael

doi:10.1016/0095-8956(91)90097-4

Cited by 144 publications

(125 citation statements)

References 3 publications

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“…For further results on dense graphs with forbidden cycles, see [2], [5], [9], [15], [17], [18], [23], [43], [46], [48].…”

Section: Examples and Applicationsmentioning

confidence: 99%

“…Many of these constructions were motivated by problems from extremal graph theory, and, as a consequence, the graphs obtained were primarily of interest in the context of a particular extremal problem. In the case of the graphs appearing in [46], [23]- [30], [16], the authors recently discovered that they exhibit many interesting properties beyond those which motivated their construction. Moreover, these properties tend to remain present even when the constructions are made far more general.…”

Section: Introductionmentioning

confidence: 99%

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General properties of some families of graphs defined by systems of equations

Lazebnik

Woldar

2001

Journal of Graph Theory

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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 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.

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“…For further results on dense graphs with forbidden cycles, see [2], [5], [9], [15], [17], [18], [23], [43], [46], [48].…”

Section: Examples and Applicationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

General properties of some families of graphs defined by systems of equations

Lazebnik

Woldar

2001

Journal of Graph Theory

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“…Bipartite graphs with high girth and their related graphs have been extensively investigated, see, e.g., [1,[8][9][10][11]13,[16][17][18]21,22,26]. We start this section with the following proposition.…”

Section: Upper Boundmentioning

confidence: 99%

New bounds on $$\bar{2}$$ 2 ¯ -separable codes of length 2

Cheng

Jiang

et al. 2013

Des. Codes Cryptogr.

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Let C be a code of length n over an alphabet of q letters. The descendant code desc(C 0 ) of C 0 = {c 1 , c 2 , . . . , c t } ⊆ C is defined to be the set of wordsThe study of separable codes is motivated by questions about multimedia fingerprinting for protecting copyrighted multimedia data. Let M(t, n, q) be the maximal possible size of such a separable code. In this paper, we provide an improved upper bound for M(2, 2, q) by a graph theoretical approach, and a new lower bound for M(2, 2, q) by deleting suitable points and lines from a projective plane, which coincides with the improved upper bound in some places. This corresponds to the bounds of maximum size of bipartite graphs with girth 6 and a construction of such maximal bipartite graphs.

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“…3 These observations immediately implied tight bounds on Ex(P 4 , n) = Θ(nα(n)) and Ex(P 5 , n) = Θ(n 3/2 ), where α is the inverse-Ackermann function. (See, e.g., [16,26,34,7]. )…”

Section: Introductionmentioning

confidence: 99%

On Nonlinear Forbidden 0–1 Matrices: A Refutation of a Füredi-Hajnal Conjecture

Pettie

2010

Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms

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A 0-1 matrix A is said to avoid a forbidden 0-1 matrix (or pattern) P if no submatrix of A matches P , where a 0 in P matches either 0 or 1 in A. The theory of forbidden matrices subsumes many extremal problems in combinatorics and graph theory such as bounding the length of Davenport-Schinzel sequences and their generalizations, Stanley and Wilf's permutation avoidance problem, and Turán-type subgraph avoidance problems. In addition, forbidden matrix theory has proved to be a powerful tool in discrete geometry and the analysis of both geometric and non-geometric algorithms.Clearly a 0-1 matrix can be interpreted as the incidence matrix of a bipartite graph in which vertices on each side of the partition are ordered. Our primary contribution is a refutation of a conjecture of Füredi and Hajnal: that if P corresponds to an acyclic graph then the maximum number of 1s in an n × n matrix avoiding P is O(n log n). In addition, we give a simpler proof that there are infinitely many minimal nonlinear patterns and give tight bounds on the extremal functions for several small forbidden patterns.

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Extremal graphs with no C4's, C6's, or C10's

Cited by 144 publications

References 3 publications

General properties of some families of graphs defined by systems of equations

General properties of some families of graphs defined by systems of equations

New bounds on $$\bar{2}$$ 2 ¯ -separable codes of length 2

On Nonlinear Forbidden 0–1 Matrices: A Refutation of a Füredi-Hajnal Conjecture

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