The theory of Dvořák et al. 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 demonstrate several properties that are necessary for every triangle-free graph embedded in the torus to be 4-critical. Most importantly we demonstrate that every such graph 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.
| Identifer | oai:union.ndltd.org:nusl.cz/oai:invenio.nusl.cz:365083 |
| Date | January 2017 |
| Creators | Pekárek, Jakub |
| Contributors | Dvořák, Zdeněk, Šámal, Robert |
| Source Sets | Czech ETDs |
| Language | English |
| Detected Language | English |
| Type | info:eu-repo/semantics/masterThesis |
| Rights | info:eu-repo/semantics/restrictedAccess |
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