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arXiv · 2609.22311

On the representation number of chessboard graphs

Sergey Kitaev·Artem Pyatkin
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Abstract

The representation number of a graph is the smallest integer $k$ such that the graph can be represented by a word in which each vertex appears exactly $k$ times, and two distinct vertices $x$ and $y$ alternate in the word if and only if they are adjacent in the graph. We extend known results on the representation number for various graph classes to chessboard graphs - namely, king, queen, rook, bishop, and knight graphs. We provide a complete classification for queen graphs and partial classifications or observations for the other classes. As a consequence of our study, we obtain a characterization of all chessboard graphs that are circle graphs. Our work also leads to several interesting open problems.

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Sergey Kitaev, Artem Pyatkin. 2026-09-15. On the representation number of chessboard graphs. https://arxiv.org/abs/2609.22311

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