### Abstract

The local chromatic number of a graph G is the number of colors appearing in the most colorful closed neighborhood of a vertex minimized over all proper colorings of G. We show that two specific topological obstructions that have the same implications for the chromatic number have different implications for the local chromatic number. These two obstructions can be formulated in terms of the homomorphism complex Hom(K _{2},G) and its suspension, respectively. These investigations follow the line of research initiated by Matoušek and Ziegler who recognized a hierarchy of the different topological expressions that can serve as lower bounds for the chromatic number of a graph. Our results imply that the local chromatic number of 4-chromatic Kneser, Schrijver, Borsuk, and generalized Mycielski graphs is 4, and more generally, that 2r-chromatic versions of these graphs have local chromatic number at least r + 2. This lower bound is tight in several cases by results of the first two authors.

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

Pages (from-to) | 889-908 |

Number of pages | 20 |

Journal | Transactions of the American Mathematical Society |

Volume | 361 |

Issue number | 2 |

DOIs | |

Publication status | Published - Feb 2009 |

### Fingerprint

### Keywords

- Borsuk-ulam theorem
- Box complex
- Local chromatic number

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Transactions of the American Mathematical Society*,

*361*(2), 889-908. https://doi.org/10.1090/S0002-9947-08-04643-6

**Local chromatic number and distinguishing the strength of topological obstructions.** / Simonyi, Gábor; Tardos, G.; Vrećica, Sinišia T.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 361, no. 2, pp. 889-908. https://doi.org/10.1090/S0002-9947-08-04643-6

}

TY - JOUR

T1 - Local chromatic number and distinguishing the strength of topological obstructions

AU - Simonyi, Gábor

AU - Tardos, G.

AU - Vrećica, Sinišia T.

PY - 2009/2

Y1 - 2009/2

N2 - The local chromatic number of a graph G is the number of colors appearing in the most colorful closed neighborhood of a vertex minimized over all proper colorings of G. We show that two specific topological obstructions that have the same implications for the chromatic number have different implications for the local chromatic number. These two obstructions can be formulated in terms of the homomorphism complex Hom(K 2,G) and its suspension, respectively. These investigations follow the line of research initiated by Matoušek and Ziegler who recognized a hierarchy of the different topological expressions that can serve as lower bounds for the chromatic number of a graph. Our results imply that the local chromatic number of 4-chromatic Kneser, Schrijver, Borsuk, and generalized Mycielski graphs is 4, and more generally, that 2r-chromatic versions of these graphs have local chromatic number at least r + 2. This lower bound is tight in several cases by results of the first two authors.

AB - The local chromatic number of a graph G is the number of colors appearing in the most colorful closed neighborhood of a vertex minimized over all proper colorings of G. We show that two specific topological obstructions that have the same implications for the chromatic number have different implications for the local chromatic number. These two obstructions can be formulated in terms of the homomorphism complex Hom(K 2,G) and its suspension, respectively. These investigations follow the line of research initiated by Matoušek and Ziegler who recognized a hierarchy of the different topological expressions that can serve as lower bounds for the chromatic number of a graph. Our results imply that the local chromatic number of 4-chromatic Kneser, Schrijver, Borsuk, and generalized Mycielski graphs is 4, and more generally, that 2r-chromatic versions of these graphs have local chromatic number at least r + 2. This lower bound is tight in several cases by results of the first two authors.

KW - Borsuk-ulam theorem

KW - Box complex

KW - Local chromatic number

UR - http://www.scopus.com/inward/record.url?scp=70449521255&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70449521255&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-08-04643-6

DO - 10.1090/S0002-9947-08-04643-6

M3 - Article

VL - 361

SP - 889

EP - 908

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 2

ER -