## Abstract

A graph homomorphism between two graphs is a map from the vertex set of one graph to the vertex set of the other graph, that maps edges to edges. In this note we study the range of a uniformly chosen homomorphism from a graph G to the infinite line ℤ. It is shown that if the maximal degree of G is ‘sub-logarithmic’, then the range of such a homomorphism is super-constant. Furthermore, some examples are provided, suggesting that perhaps for graphs with superlogarithmic degree, the range of a typical homomorphism is bounded. In particular, a sharp transition is shown for a specific family of graphs C_{n,k} (which is the tensor product of the n-cycle and a complete graph, with self-loops, of size k). That is, given any function ψ(n) tending to infinity, the range of a typical homomorphism of C_{n,k} is super-constant for k = 2 log(n) − ψ(n), and is 3 for k = 2 log(n) + ψ(n).

Original language | English |
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Pages (from-to) | 926-950 |

Number of pages | 25 |

Journal | Electronic Journal of Probability |

Volume | 12 |

DOIs | |

State | Published - 1 Jan 2007 |

Externally published | Yes |

## Keywords

- Graph Homomorphisms
- Graph Indexed Random Walks

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty