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arXiv · 2106.01037

Note on $4$-coloring $6$-regular triangulations on the torus

Brahadeesh Sankarnarayanan
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Abstract

In 1973, Altshuler characterized the $6$-regular triangulations on the torus to be precisely those that are obtained from a regular triangulation of the $r \times s$ toroidal grid where the vertices in the first and last column are connected by a shift of $t$ vertices. Such a graph is denoted $T(r, s, t)$. In 1999, Collins and Hutchinson classified the $4$-colorable graphs $T(r, s, t)$ with $r, s \geq 3$. In this paper, we point out a gap in their classification and show how it can be fixed. Combined with the classification of the $4$-colorable graphs $T(1, s, t)$ by Yeh and Zhu in 2003, this completes the characterization of the colorability of all the $6$-regular triangulations on the torus.

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Brahadeesh Sankarnarayanan. 2021-06-02. Note on $4$-coloring $6$-regular triangulations on the torus. https://doi.org/10.1007/s00026-022-00573-8

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