Bounds on upper transversals in hypergraphs

Michael A. Henning () and Anders Yeo ()
Additional contact information
Michael A. Henning: University of Johannesburg
Anders Yeo: University of Johannesburg

Journal of Combinatorial Optimization, 2020, vol. 39, issue 1, No 6, 77-89

Abstract: Abstract A set S of vertices in a hypergraph H is a transversal if it has a nonempty intersection with every edge of H. For $$k \ge 1$$k≥1, if H is a hypergraph with every edge of size at least k, then a k-transversal in H is a transversal that intersects every edge of H in at least k vertices. In particular, a 1-transversal is a transversal. The upper k-transversal number $$\Upsilon _{k}(H)$$Υk(H) of H is the maximum cardinality of a minimal k-transversal in H. Let H be a hypergraph with $$n_{_H}$$nH vertices and $$m_{_H}$$mH edges. We show that for $$r \ge 2$$r≥2 and for every integer $$k \in [r]$$k∈[r], if H is r-uniform with maximum degree $$\Delta $$Δ, then $$\Upsilon _{k}(H) \le \left( \frac{k \cdot \Delta }{k (\Delta - 1) + r} \right) n_{_H}$$Υk(H)≤k·Δk(Δ-1)+rnH and $$\Upsilon _{k}(H) \le \left( \frac{k \cdot \Delta }{\Delta (k + 1) + r - k} \right) (n_{_H}+ m_{_H})$$Υk(H)≤k·ΔΔ(k+1)+r-k(nH+mH), and both bounds are tight. As a special case of this result, if H is a 3-regular, 3-uniform hypergraph, then $$\Upsilon _{2}(H) \le \frac{6}{7} n_{_H}$$Υ2(H)≤67nH, and equality in this bound is achieved by the Fano plane. We also discuss a relation between upper transversals in 3-uniform hypergraphs and the famous cap set problem, and show that for every given $$\epsilon > 0$$ϵ>0, there exists a 3-uniform, connected, linear hypergraphs of sufficiently large order such that $$\Upsilon _{1}(H)

Keywords: Transversal; Upper transversal; k-transversal number; 05C69 (search for similar items in EconPapers)
Date: 2020
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10878-019-00456-4 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jcomop:v:39:y:2020:i:1:d:10.1007_s10878-019-00456-4

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/10878

DOI: 10.1007/s10878-019-00456-4

Access Statistics for this article

Journal of Combinatorial Optimization is currently edited by Thai, My T.

More articles in Journal of Combinatorial Optimization from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().