Tight description of faces of triangulations on the torus

The degree d(x) of a vertex or face x in a graph G is the number of incident edges. A face f=v1…vd(f) in a graph G on the plane or other orientable surface is of type (k1,k2,…), where k1≤k2≤…, if d(vi)≤ki for each i. By δ we denote the minimum vertex-degree of G. In 1989, Borodin confirmed Kotzig's conjecture of 1963 that every plane graph with minimum degree δ equal to 5 has a (5,5,7)-face or a (5,6,6)-face, where all parameters are tight. Recently, we proved that every torus triangulation with δ≥5 has a face of one of the types (5,5,8), (5,6,7), or (6,6,6), which is tight. It follows from the classical theorem by Lebesgue (1940) that every plane triangulation with δ≥4 has a 3-face of types (4,4,∞), (4,5,19), (4,6,11), (4,7,9), (5,5,9), or (5,6,7). In 1999, Jendrol' gave a similar description: “(4,4,∞), (4,5,13), (4,6,17), (4,7,8), (5,5,7), (5,6,6)” and conjectured that “(4,4,∞), (4,5,10), (4,6,15), (4,7,7), (5,5,7), (5,6,6)” holds. In 2002, Lebesgue's description was strengthened by Borodin to “(4,4,∞), (4,5,17), (4,6,11), (4,7,8), (5,5,8), (5,6,6)”. In 2014, Borodin and Ivanova obtained the following tight description, which, in particular, disproves the above mentioned conjecture by Jendrol': “(4,4,∞), (4,5,11), (4,6,10), (4,7,7), (5,5,7), (5,6,6)”, and recently proved another tight description of faces in plane triangulations with δ≥4: “(4,4,∞), (4,6,10), (4,7,7), (5,5,8), (5,6,7)”. It follows from Lebesgue's theorem of 1940 that every plane quadrangulation with δ≥3 has a face of one of the types (3,3,3,∞), (3,3,4,11), (3,3,5,7), (3,4,4,5). Recently, Borodin and Ivanova improved this description to “(3,3,3,∞), (3,3,4,9), (3,3,5,6), (3,4,4,5)”, where all parameters except possibly 9 are best possible and 9 cannot go down below 8. In 1995, Avgustinovich and Borodin proved the following tight description of the faces of torus quadrangulations with δ≥3: “(3,3,3,∞), (3,3,4,10), (3,3,5,7), (3,3,6,6), (3,4,4,6), (4,4,4,4)”, which also holds for each higher surface provided that its quadrangulation is large enough. Recently, Borodin and Ivanova proved that every triangulation with δ≥4 of the torus has a face of one of the types (4,4,∞), (4,6,12), (4,8,8), (5,5,8), (5,6,7), or (6,6,6), which description is tight. The purpose of this paper is to prove that every triangulation with δ≥3 on the torus has a face of one of the types (3,6,24), (3,8,16), (3,12,12), (4,4,∞), (4,6,12), (4,8,8), (5,5,8), (5,6,7), or (6,6,6), where all parameters are best possible.