Hypergraphs: Combinatorics of Finite Sets
Elsevier, 1 may 1984 - 254 páginas
Graph Theory has proved to be an extremely useful tool for solving combinatorial problems in such diverse areas as Geometry, Algebra, Number Theory, Topology, Operations Research and Optimization. It is natural to attempt to generalise the concept of a graph, in order to attack additional combinatorial problems. The idea of looking at a family of sets from this standpoint took shape around 1960. In regarding each set as a ``generalised edge'' and in calling the family itself a ``hypergraph'', the initial idea was to try to extend certain classical results of Graph Theory such as the theorems of Turán and König. It was noticed that this generalisation often led to simplification; moreover, one single statement, sometimes remarkably simple, could unify several theorems on graphs. This book presents what seems to be the most significant work on hypergraphs.
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admits Application arcs balanced hypergraph bicolouring bipartite called cardinality Chapter chromatic number classes Clearly cliques colour columns complete components condition connected consider contains contradiction Corollary cover defined denote determine disjoint distinct dual edges of H elements equality equivalent Erdös Example Exercise exists extreme Figure fractional Further given gives graph G H satisfies Helly property Hence holds hypergraph of order implies incidence independent set induction inequality integer interval joining k-colouring König property least Lemma length Let G Let H linear loops Math matrix matroid maximal maximum maximum matching meets Mengerian minimal necessary obtain odd cycle optimal pair partition path plane points possible problem Proof Proposition proved r-uniform hypergraph rank representative result rows satisfies shown simple hypergraph strongly subhypergraph subsets suppose Theorem Theory Tr H transversal tree uniform unimodular vector vertex vertices weight whence
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