Note on 4-Coloring 6-Regular Triangulations on the Torus

Altshuler (Discrete Math 4(3):201–217, 1973) characterized the 6-regular triangulations on the torus to be precisely those that are obtained from a regular triangulation of the r× 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). Collins and Hutchinson (Graph colouring and applications. CRM proceedings and lecture notes, vol 23. American Mathematical Society, Providence, pp 21–34, 1999) classified the 4-colorable graphs T(r, s, t) with r, s≥ 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 (Discrete Math 273(1–3):261–274, 2003), this completes the characterization of the colorability of all the 6-regular triangulations on the torus.