| author: | Matěj Stehlík |
|---|---|
| title: |
Connected
τ
-critical hypergraphs of minimal size
|
| keywords: |
τ
-critical hypergraph,
χ
-critical
3
-chromatic hypergraph
|
| abstract: |
A hypergraph
H
τ
-critical if
τ(
for every edge
H
-E) < τ(
H
)
E ∈
, where
H
τ(
denotes the transversal number of
H
)
H
τ
-critical hypergraph
H
2τ(
edges; this generalises a classical theorem of Gallai
on
H
)-1
χ
-vertex-critical graphs with connected complements.
In this paper we study connected
τ
-critical hypergraphs
H
2τ(
edges. We prove that such hypergraphs have at least
H
)-1
2τ(
vertices, and characterise those with
H
)-1
2τ(
vertices using a directed odd ear decomposition of an
associated digraph. Using Seymour's characterisation of
H
)-1
χ
-critical
3
-chromatic square hypergraphs, we also show that a
connected square hypergraph
H
2τ(
edges is
H
)
τ
-critical if and only if it is
χ
-critical
3
-chromatic. Finally, we deduce some new results on
χ
-vertex-critical graphs with connected complements.
|
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| reference: |
Matěj Stehlík (2005), Connected
τ
-critical hypergraphs of minimal size, in 2005
European Conference on Combinatorics, Graph Theory and
Applications (EuroComb '05), Stefan Felsner (ed.),
Discrete Mathematics and Theoretical Computer Science
Proceedings AE, pp. 157-160
|
| bibtex: | For a corresponding BibTeX entry, please consider our BibTeX-file. |
| ps.gz-source: | dmAE0131.ps.gz (215 K) |
| ps-source: | dmAE0131.ps (395 K) |
| pdf-source: | dmAE0131.pdf (140 K) |
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