Connected τ -critical hypergraphs of minimal size
A hypergraph $\mathscr{H}$ is $τ$ -critical if $τ (\mathscr{H}-E) < τ (\mathscr{H})$ for every edge $E ∈\mathscr{H}$, where $τ (\mathscr{H})$ denotes the transversal number of $\mathscr{H}$. It can be shown that a connected $τ$ -critical hypergraph $\mathscr{H}$ has at least $2τ (\mathscr{H})-1$...
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| Format: | Article |
| Language: | English |
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Discrete Mathematics & Theoretical Computer Science
2005-01-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
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| Online Access: | https://dmtcs.episciences.org/3397/pdf |
| _version_ | 1797270393526943744 |
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| author | Matěj Stehlík |
| author_facet | Matěj Stehlík |
| author_sort | Matěj Stehlík |
| collection | DOAJ |
| description | A hypergraph $\mathscr{H}$ is $τ$ -critical if $τ (\mathscr{H}-E) < τ (\mathscr{H})$ for every edge $E ∈\mathscr{H}$, where $τ (\mathscr{H})$ denotes the transversal number of $\mathscr{H}$. It can be shown that a connected $τ$ -critical hypergraph $\mathscr{H}$ has at least $2τ (\mathscr{H})-1$ edges; this generalises a classical theorem of Gallai on $χ$ -vertex-critical graphs with connected complements. In this paper we study connected $τ$ -critical hypergraphs $\mathscr{H}$ with exactly $2τ (\mathscr{H)}-1$ edges. We prove that such hypergraphs have at least $2τ (\mathscr{H})-1$ vertices, and characterise those with $2τ (\mathscr{H})-1$ vertices using a directed odd ear decomposition of an associated digraph. Using Seymour's characterisation of $χ$ -critical 3-chromatic square hypergraphs, we also show that a connected square hypergraph $\mathscr{H}$ with fewer than $2τ (\mathscr{H})$ edges is $τ$ -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. |
| first_indexed | 2024-04-25T02:03:33Z |
| format | Article |
| id | doaj.art-fe7c9618e8b6494488f37f383b4f8441 |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:03:33Z |
| publishDate | 2005-01-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-fe7c9618e8b6494488f37f383b4f84412024-03-07T14:41:15ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502005-01-01DMTCS Proceedings vol. AE,...Proceedings10.46298/dmtcs.33973397Connected τ -critical hypergraphs of minimal sizeMatěj Stehlík0Universidad Nacional Autónoma de México = National Autonomous University of MexicoA hypergraph $\mathscr{H}$ is $τ$ -critical if $τ (\mathscr{H}-E) < τ (\mathscr{H})$ for every edge $E ∈\mathscr{H}$, where $τ (\mathscr{H})$ denotes the transversal number of $\mathscr{H}$. It can be shown that a connected $τ$ -critical hypergraph $\mathscr{H}$ has at least $2τ (\mathscr{H})-1$ edges; this generalises a classical theorem of Gallai on $χ$ -vertex-critical graphs with connected complements. In this paper we study connected $τ$ -critical hypergraphs $\mathscr{H}$ with exactly $2τ (\mathscr{H)}-1$ edges. We prove that such hypergraphs have at least $2τ (\mathscr{H})-1$ vertices, and characterise those with $2τ (\mathscr{H})-1$ vertices using a directed odd ear decomposition of an associated digraph. Using Seymour's characterisation of $χ$ -critical 3-chromatic square hypergraphs, we also show that a connected square hypergraph $\mathscr{H}$ with fewer than $2τ (\mathscr{H})$ edges is $τ$ -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.https://dmtcs.episciences.org/3397/pdfτ -critical hypergraphχ -critical 3-chromatic hypergraph[info.info-dm] computer science [cs]/discrete mathematics [cs.dm][math.math-co] mathematics [math]/combinatorics [math.co] |
| spellingShingle | Matěj Stehlík Connected τ -critical hypergraphs of minimal size Discrete Mathematics & Theoretical Computer Science τ -critical hypergraph χ -critical 3-chromatic hypergraph [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] [math.math-co] mathematics [math]/combinatorics [math.co] |
| title | Connected τ -critical hypergraphs of minimal size |
| title_full | Connected τ -critical hypergraphs of minimal size |
| title_fullStr | Connected τ -critical hypergraphs of minimal size |
| title_full_unstemmed | Connected τ -critical hypergraphs of minimal size |
| title_short | Connected τ -critical hypergraphs of minimal size |
| title_sort | connected τ critical hypergraphs of minimal size |
| topic | τ -critical hypergraph χ -critical 3-chromatic hypergraph [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] [math.math-co] mathematics [math]/combinatorics [math.co] |
| url | https://dmtcs.episciences.org/3397/pdf |
| work_keys_str_mv | AT matejstehlik connectedtcriticalhypergraphsofminimalsize |