# The k-dimensional cube is k-representable

Bas Broere, Hans Zantema

### 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 3-12 10 Journal of Automata, Languages and Combinatorics 24 1 https://doi.org/10.25596/jalc-2019-003 Published - 2019

### Fingerprint

Regular hexahedron
Vertex of a graph
Concatenation
Cartesian product
Graph in graph theory
Complete Graph
Alternate
Strings
Distinct
Theorem

### Keywords

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

### Cite this

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title = "The k-dimensional cube is k-representable",
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.",
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author = "Bas Broere and Hans Zantema",
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language = "English",
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In: Journal of Automata, Languages and Combinatorics, Vol. 24, No. 1, 2019, p. 3-12.

TY - JOUR

T1 - The k-dimensional cube is k-representable

AU - Broere, Bas

AU - Zantema, Hans

PY - 2019

Y1 - 2019

N2 - 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.

AB - 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.

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DO - 10.25596/jalc-2019-003

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