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arXiv · 2311.10379

Achromatic colorings of polarity graphs

Vladislav Taranchuk·Craig Timmons
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Abstract

A complete partition of a graph $G$ is a partition of the vertex set such that there is at least one edge between any two parts. The largest $r$ such that $G$ has a complete partition into $r$ parts, each of which is an independent set, is the achromatic number of $G$. We determine the achromatic number of polarity graphs of biaffine planes coming from generalized polygons. Our colorings of a family of unitary polarity graphs are used to solve a problem of Axenovich and Martin on complete partitions of $C_4$-free graphs. Furthermore, these colorings prove that there are sequences of graphs which are optimally complete and have unbounded degree, a problem that had been studied for the sequence of hypercubes independently by Roichman, and Ahlswede, Bezrukov, Blokhuis, Metsch, and Moorhouse.

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Vladislav Taranchuk, Craig Timmons. 2023-11-17. Achromatic colorings of polarity graphs. https://arxiv.org/abs/2311.10379

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