Abstract
The theory of Dvorak, Kral', and Thomas (Dvorak, 2015) shows that a 4-critical triangle-free graph embedded in the torus has only a bounded number of faces of length greater than 4 and that the size of these faces is also bounded. We study the natural reduction in such embedded graphs-identification of opposite vertices in 4-faces.
We give a computer-assisted argument showing that there are exactly four 4-critical triangle-free irreducible toroidal graphs in which this reduction cannot be applied without creating a triangle. Using this result, we show that every 4-critical triangle-free graph embedded in the torus has at most four 5-faces, or a 6-face and two 5-faces, or a 7face and a 5-face, in addition to at least seven 4-faces.
This result serves as a basis for the exact description of 4-critical triangle-free toroidal graphs, which we present in a followup paper. (C) 2020 Elsevier Ltd. All rights reserved.