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

Altshuler (Discrete Math 4(3):201–217, 1973) characterized the 6-regular triangulations on torus to be precisely those that are obtained from a regular triangulation of $$r \times s$$ toroidal grid where vertices in first and last column connected by shift t vertices. Such graph is denoted T(r, s, t). Collins Hutchinson (Graph colouring applications. CRM proceedings lecture notes, vol 23. American Mathematical Society, Providence, pp 21–34, 1999) classified 4-colorable graphs t) with $$r, s \ge 3$$ . In this paper, we point out gap their classification show how it can fixed. Combined T(1, Yeh Zhu 273(1–3):261–274, 2003), completes characterization colorability all torus.