Abstract
A hypergraph is τ-critical if τ(−{E})<τ() for every edge E ∈ , where τ() denotes the transversal number of . We show that if is a connected τ-critical hypergraph, then −{E} can be partitioned into τ()−1 stars of size at least two, for every edge E ∈ . An immediate corollary is that a connected τ-critical hypergraph has at least 2τ()−1 edges. This extends, in a very natural way, a classical theorem of Gallai on colour-critical graphs, and is equivalent to a theorem of Füredi on t-stable hypergraphs. We deduce a lower bound on the size of τ-critical hypergraphs of minimum degree at least two.
Similar content being viewed by others
References
Berge, C.: Hypergraphs: Combinatorics of Finite Sets. North-Holland, Amsterdam, 1989
Duchet, P.: Hypergraphs. In: R.L. Graham et al.: Handbook of Combinatorics, vol. 1. North-Holland, Amsterdam, 1995, pp. 381–432
Frankl, P.: Extremal set systems. In: R.L. Graham et al.: Handbook of Combinatorics, vol. 2. North-Holland, Amsterdam, 1995, pp. 1293–1329
Füredi, Z.: t-Expansive and t-wise intersecting hypergraphs. Graphs Combin. 2, 67–80 (1986)
Füredi, Z.: Matchings and covers in hypergraphs. Graphs Combin. 4, 115–206 (1988)
Gallai, T.: Neuer Beweis eines Tutte'schen Satzes. Mag. Tud. Akad. Kutató Int. Közl. 8, 135–139 (1963)
Gallai, T.: Kritische Graphen II. Mag. Tud. Akad. Kutató Int. Közl. 8, 373–395 (1963)
Molloy, M.: Chromatic neighborhood sets. J. Graph Theory 31, 303–311 (1999)
Stehlík, M.: Critical graphs with connected complements. J. Combin. Theory Ser. B 89, 189–194 (2003)
Stehlík, M.: Connected τ-critical hypergraphs of minimal size. In: Felsner, S.: EuroComb '05, Berlin 2005 (Discrete Math. Theor. Comput. Sci., Proc., vol. AE, DMTCS, Nancy 2005, pp. 157–160
Stehlík, M.: On the structure of minimal connected τ-critical hypergraphs (In preparation)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Stehlík, M. Minimal Connected τ-Critical Hypergraphs. Graphs and Combinatorics 22, 421–426 (2006). https://doi.org/10.1007/s00373-006-0656-1
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s00373-006-0656-1