The k-cube is k-representable

TL;DR

This paper proves that the k-cube graph is k-representable for all k ≥ 1 by constructing specific words based on occurrence functions, expanding understanding of graph representability through combinatorial word properties.

Contribution

It establishes that the Cartesian product of a k-representable graph with a complete graph results in a (k+n-1)-representable graph, demonstrating the k-representability of the k-cube.

Findings

The k-cube is k-representable for all k ≥ 1.

A new technique using occurrence-based functions is introduced.

Constructed representing words are composed of concatenation and occurrence functions.

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 $k$-cube is $k$-representable for every $k \geq 1$. Our main technique consists of exploring occurrence based functions that replace every $i$th 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.