### Abstract

CWI, Kruislaan 413, 1098 SJ, Amsterdam, The Netherlands Let W ⊆ V in a graph G = (V, E) such that W ∩ X ≠ ∅ for each fragment X of G. Then G is defined to be W-locally (k, l)-critical if κ(G - W′) = k - |W′| holds for every W′ ⊆ W with |W′| ≤ l. In this note we give a short proof for the following recent result of Su: every non-complete W-locally (k, l)-critical graph has (2l + 2) distinct ends and |W| ≥ 2l + 2. (This result implies that Slater's conjecture is true: there exist no (k, l)-critical graphs with 2l > k, except K_{k + 1.}).

Original language | English |
---|---|

Pages (from-to) | 273-275 |

Number of pages | 3 |

Journal | Discrete Mathematics |

Volume | 179 |

Issue number | 1-3 |

Publication status | Published - Jan 15 1998 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

*179*(1-3), 273-275.

**On the existence of (k, l)-critical graphs.** / Jordán, T.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 179, no. 1-3, pp. 273-275.

}

TY - JOUR

T1 - On the existence of (k, l)-critical graphs

AU - Jordán, T.

PY - 1998/1/15

Y1 - 1998/1/15

N2 - CWI, Kruislaan 413, 1098 SJ, Amsterdam, The Netherlands Let W ⊆ V in a graph G = (V, E) such that W ∩ X ≠ ∅ for each fragment X of G. Then G is defined to be W-locally (k, l)-critical if κ(G - W′) = k - |W′| holds for every W′ ⊆ W with |W′| ≤ l. In this note we give a short proof for the following recent result of Su: every non-complete W-locally (k, l)-critical graph has (2l + 2) distinct ends and |W| ≥ 2l + 2. (This result implies that Slater's conjecture is true: there exist no (k, l)-critical graphs with 2l > k, except Kk + 1.).

AB - CWI, Kruislaan 413, 1098 SJ, Amsterdam, The Netherlands Let W ⊆ V in a graph G = (V, E) such that W ∩ X ≠ ∅ for each fragment X of G. Then G is defined to be W-locally (k, l)-critical if κ(G - W′) = k - |W′| holds for every W′ ⊆ W with |W′| ≤ l. In this note we give a short proof for the following recent result of Su: every non-complete W-locally (k, l)-critical graph has (2l + 2) distinct ends and |W| ≥ 2l + 2. (This result implies that Slater's conjecture is true: there exist no (k, l)-critical graphs with 2l > k, except Kk + 1.).

UR - http://www.scopus.com/inward/record.url?scp=0013019861&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0013019861&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0013019861

VL - 179

SP - 273

EP - 275

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 1-3

ER -