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- W2019480324 abstract "Let S be a set and let { X 1 , …, X n } = be a family of distinct subsets of S . The intersection graph Ω( ) of has vertex set { X 1 , …, X n } and X i X j ( i ≠ j ) is an edge of Ω( ) if and only if X i ∩ X i ≠ Ø (c.f. (6)). It is easily seen, (7), that every graph is an intersection graph, in other words every graph can be represented by subsets ofa set. Moreover, it was proved by Erdös, Goodman and Pósa (5) that if a graph has n ≥ 4 vertices then one can find a representing set with at most [ n 2 /4] elements. This assertion is an immediate consequence of the result, (5), that the edges of a graph with n ≥ 1 vertices can be covered with at most [ n 2 /4] edge disjoint triangles and edges. We say that a graph G is covered with the subgraphs G 1 , …, G k , if every edge of G is in at least one G i . One of the aims of this note is to prove an extension of this result, pro-posed by Erdös (4)." @default.
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- W2019480324 date "1976-01-01" @default.
- W2019480324 modified "2023-10-17" @default.
- W2019480324 title "On complete subgraphs of different orders" @default.
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- W2019480324 doi "https://doi.org/10.1017/s0305004100052063" @default.
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