### Abstract

A graph is called k-representable if there exists a word w over the nodes of the graph, each node occurring exactly k times, such that there is an edge between two nodes x, y if and only after removing all letters distinct from x, y, from w, a word remains in which x, y alternate. We prove that if G is k-representable for k > 1, then the Cartesian product of G and the complete graph on n nodes is (k + n − 1)-representable. As a direct consequence, the k-dimensional cube is k-representable for every k ≥ 1. Our main technique consists of exploring occurrence-based functions that replace every ith occurrence of a symbol x in a word w by a string h(x, i). The representing word we construct to achieve our main theorem is purely composed from concatenation and occurrence-based functions.

Original language | English |
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Pages (from-to) | 3-12 |

Number of pages | 10 |

Journal | Journal of Automata, Languages and Combinatorics |

Volume | 24 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2019 |

### Keywords

- Cartesian product graph
- K-dimensional cube
- Word representation

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## Cite this

*Journal of Automata, Languages and Combinatorics*,

*24*(1), 3-12. https://doi.org/10.25596/jalc-2019-003