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

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Rucinski, A., Vince, A. Balanced extensions of graphs and hypergraphs. Combinatorica 8, 279–291 (1988). https://doi.org/10.1007/BF02126800

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AMS subject classification (1980)

  • 05 C 35
  • 05 C 65