Balanced extensions of graphs and hypergraphs


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.

This is a preview of subscription content, access via your institution.


  1. [1]

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

    Google Scholar 

  2. [2]

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

  3. [3]

    M. Karoński,Balanced Subgraphs of Large Random Graphs, A. M. Univ. Press, Poznań, 1984.

    Google Scholar 

  4. [4]

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

  5. [5]

    A.Ruciński and A.Vince, Balanced graphs and the problems of subgraphs of random graphs,submitted.

Download references

Author information



Additional information

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

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Rucinski, A., Vince, A. Balanced extensions of graphs and hypergraphs. Combinatorica 8, 279–291 (1988).

Download citation

AMS subject classification (1980)

  • 05 C 35
  • 05 C 65