Graph Theory

Hypergraphs: combinatorics of finite sets by C. Berge

By C. Berge

Graph concept has proved to be a very useful gizmo for fixing combinatorial difficulties in such diversified components as Geometry, Algebra, quantity concept, Topology, Operations learn and Optimization. it really is normal to try to generalise the idea that of a graph, with a purpose to assault extra combinatorial difficulties. the belief of taking a look at a relations of units from this viewpoint took form round 1960. In relating to each one set as a ``generalised edge'' and in calling the relations itself a ``hypergraph'', the preliminary concept used to be to attempt to increase yes classical result of Graph idea comparable to the theorems of Turán and König. It was once spotted that this generalisation frequently ended in simplification; furthermore, one unmarried assertion, occasionally remarkably basic, may unify numerous theorems on graphs. This e-book offers what seems the main major paintings on hypergraphs.

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Extra info for Hypergraphs: combinatorics of finite sets

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If a is the largest integer such that m ( H ) 2 (f) then 4 W I k ) 1);( Proof. Let H I be a partial hypergraph of H with m ( H l ) = (:). Q2)9 m(~2Ir-2) etc. Since [HI, 3 [Hr-& we also have L (;I. D. 7. Conformal Hypergraphs We say that a hypergraph H is conformal if all the maximal cliques of the graph [HI2 are edges of H . If H is simple, it is conformal if and only if the edges of H are the maximal cliques of a graph. More generally, consider an integer k 2 2. Every edge A of a hypergraph H satisfies the property: the edges of [HI, contained in A constitute a k-complete hypergraph.

By an a f f i n e plane is meant the subhypergraph H of rank k obtained from a finite projective plane of rank k + l by suppressing the points of a given line. Every edge of H is called a line, and two lines of H which have an empty intersection are said to be parallel. Thus an affine plane of rank k satisfies the following properties: Every line contains k points; Every point belongs to k + l lines; There are k 2 points and k2 k lines; Two distinct points have one and only one line in common; Two distinct lines have either no points in common (“parallel”), or a common point (“secant”); Parallelism is an equivalence relation which partitions the set of lines into k + l classes of k edges each; Through every point not belonging to a given line, there passes one and only one line parallel to the given line.

X-Em) is a simple hypergraph. Deduce the following inequality (SchGnheim [lQSS]): and this bound is best possible. ,Am) , be a family of m circular intervals on a circle of n points with (ii) I>n/2; A; f l Ag # 0 (iii) A, &Aj (i) IAi (i#j) (i+j) 5 n , with equality if n having fixed cardinality k > -. 2 Then we have m A is the family Ak of distinct circular intervals Exercise 8 ($3) Let A be a family of circular intervals satisfying conditions (2) and (3) of Exercise 7, and for A € A , put: Show that Cp(A) 5 n .

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