Irreducible 4-critical triangle-free toroidal graphs. (August 2020)

• Record Type: Journal Article
• Authors: Dvořák, Zdeněk; Pekárek, Jakub
• Is Part Of: European journal of combinatorics. Volume 88(2020)
• Publication Date: 2020-08
• Subjects: Combinatorial analysis -- Periodicals; Analyse combinatoire -- Périodiques; Combinatorial analysis; Periodicals; Electronic journals; 511.6
• DOI: 10.1016/j.ejc.2020.103112
• Languages: English
• ISSNs: 0195-6698

Abstract

Abstract: The theory of Dvořák, Král', and Thomas (Dvořák, 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 7-face 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.