### Abstract

Given an infinite graph G, let deg_{∞}(G) be defined as the smallest d for which V(G) can be partitioned into finite subsets of (uniformly) bounded size such that each part is adjacent to at most d others. A countable graph G is constructed with deg_{∞}(G)>2 and with the property that |{yε{lunate}V(G):d(x,y)≤n}|≤Cn for any xε{lunate}V(G), nε{lunate}N. This disproves conjectures of Cenzer and Howorka.

Original language | English (US) |
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Pages (from-to) | 107-108 |

Number of pages | 2 |

Journal | Discrete Mathematics |

Volume | 79 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 1990 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

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## Cite this

Pach, J., & Spencer, J. H. (1990). Note on vertex-partitions of infinite graphs.

*Discrete Mathematics*,*79*(1), 107-108. https://doi.org/10.1016/0012-365X(90)90060-U