# Balanced extensions of graphs and hypergraphs

## Abstract

For a hypergraphG withv vertices ande i edges of sizei, the average vertex degree isd(G)= ∑ie 1/v. Callbalanced ifd(H)≦d(G) for all subhypergraphsH ofG. Let

$$m(G) = \mathop {\max }\limits_{H \subseteqq G} d(H).$$

A hypergraphF is said to be abalanced extension ofG ifG⊂F, F is balanced andd(F)=m(G), i.e.F is balanced and does not increase the maximum average degree. It is shown that for every hypergraphG there exists a balanced extensionF ofG. Moreover everyr-uniform hypergraph has anr-uniform balanced extension. For a graphG let ext (G) denote the minimum number of vertices in any graph that is a balanced extension ofG. IfG hasn vertices, then an upper bound of the form ext(G)<c 1 n 2 is proved. This is best possible in the sense that ext(G)>c 2 n 2 for an infinite family of graphs. However for sufficiently dense graphs an improved upper bound ext(G)<c 3 n can be obtained, confirming a conjecture of P. Erdõs.

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## References

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P. Erdős andA. Rényi, On the evolution of random graphs,Publ. Math. Inst. Hung. Acad. Sci. 5 (1960), 17–61.

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E.Győri, B.Rothschild and A.Ruciński, Every graph is contained in a sparest possible balanced graph,submitted.

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A.Ruciński and A.Vince, Strongly balanced graphs and random graphs,submitted.

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## Author information

### Affiliations

Authors

On leave from Institute of Mathematics, Adam Mickiewicz University, Poznan, Poland

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