A subset of the vertices in a hypergraph is a cover if it intersects every edge. Let τ(H) denote the cardinality of a minimum cover in the hypergraph H, and let us denote by g(n) the maximum of τ(H) taken over all hypergraphs H with n vertices and with no two hyperedges of the same size. We show thatg(n)<1.98n(1+o(1)). A special case corresponds to an old problem of Erdos asking for the maximum number of edges in an n-vertex graph with no two cycles of the same length. Denoting this maximum by n+f(n), we can show that f(n)≤1.98n(1+o(1)). Generalizing the above, let g(n, C, k) denote the maximum of τ(H) taken over all hypergraph H with n vertices and with at most Cik edges with cardinality i for all i=1, 2, ..., n. We prove that g(n, C, k)<(Ck!+1)n(k+1)/(k+2).These results have an interesting graph-theoretic application. For a family F of graphs, let T(n, F, r) denote the maximum possible number of edges in a graph with n vertices, which contains each member of F at most r-1 times. T(n, F, 1)=T(n, F) is the classical Turán number. Using the results above, we can compute a non-trivial upper bound for T(n, F, r) for many interesting graph families.
Bibliographical noteFunding Information:
The authors are indebted to Janice Malouf for fruitful discussions. The first author also gratefully acknowledges the partial support by the Office of Naval Research (Grant N00014-92-J-1375) and by the National Science Foundation (Grant DMS 98-06389).
1Research supported in part by the Hungarian National Science Foundation under Grant OTKA 016389 and by National Security Agency Grant MDA904-98-1-0022.
- Hypergraph; covering; cycles
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics