## Abstract

A path system (Formula presented.) in a graph (Formula presented.) is a collection of paths with a unique (Formula presented.) path for every two vertices (Formula presented.). We say that (Formula presented.) is consistent if for any path (Formula presented.), every subpath of (Formula presented.) is also in (Formula presented.). It is metrizable if there exists a positive weight function (Formula presented.) such that (Formula presented.) is comprised of (Formula presented.) -shortest paths. We call (Formula presented.) irreducible if there does not exist a partition (Formula presented.) such that (Formula presented.) restricts to a path system on both (Formula presented.) and (Formula presented.). In this paper, we construct an infinite family of nonmetrizable irreducible consistent path systems on certain Paley graphs.

Original language | American English |
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Pages (from-to) | 5-14 |

Number of pages | 10 |

Journal | Journal of Graph Theory |

Volume | 102 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2023 |

### Bibliographical note

Funding Information:Open access publishing facilitated by MALMAD hybrid - Hebrew University of Jerusalem.

Publisher Copyright:

© 2022 The Authors. Journal of Graph Theory published by Wiley Periodicals LLC.

## Keywords

- Paley graphs
- irreducibility
- metrizability
- path systems