2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)
Stefan Felsner (ed.)
DMTCS Conference Volume AE (2005), pp. 157160
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
vertexcritical 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
vertexcritical 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. 157160

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