Uniquely K r (K) -saturated hypergraphs

A. Gyárfás, Stephen G. Hartke, Charles Viss

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Abstract

In this paper we generalize the concept of uniquely K r -saturated graphs to hypergraphs. Let K r (k) denote the complete k-uniform hypergraph on r vertices. For integers k, r, n such that 2  k < r < n, a k-uniform hypergraph H with n vertices is uniquely K r (k) -saturated if H does not contain K r (k) but adding to H any k-set that is not a hyperedge of H results in exactly one copy of K r (k) . Among uniquely K r (k) -saturated hypergraphs, the interesting ones are the primitive( k−1 )ones that do not have a dominating vertex—a vertex belonging to all possible (Formula presented) edges. Translating the concept to the complements of these hypergraphs, we obtain a natural restriction of τ-critical hypergraphs: a hypergraph H is uniquely τ-critical if for every edge e, τ (H − e) = τ (H) − 1 and H − e has a unique transversal of size τ (H) − 1. We have two constructions for primitive uniquely K r (k) -saturated hypergraphs. One shows that for k and r where 4  k < r  2k−3, there exists such a hypergraph for every n > r. This is in contrast to the case k = 2 and r = 3 where only the Moore graphs of diameter two have this property. Our other construction keeps n − r fixed; in this case we show that for any fixed k  2 there can only be finitely many examples. We give a range for n where these hypergraphs exist. For n−r = 1 the range is completely determined: k + 1  n (Formula presented). For larger values of n − r the upper end of our range reaches approximately half of its upper bound. The lower end depends on the chromatic number of certain Johnson graphs.

Original languageEnglish
Article number#P4.35
JournalElectronic Journal of Combinatorics
Volume25
Issue number4
Publication statusPublished - Jan 1 2018

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ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics
  • Applied Mathematics

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