Three-coloring triangle-free graphs on surfaces III. Graphs of girth five

We show that the size of a 4-critical graph of girth at least five is bounded by a linear function of its genus. This strengthens the previous bound on the size of such graphs given by Thomassen. It also serves as the basic case for the description of the structure of 4-critical triangle-free graphs embedded in a fixed surface, presented in a future paper of this series.Comment: 53 pages, 7 figures; updated according to referee remark

Three-coloring triangle-free graphs on surfaces II. 4-critical graphs in a disk

Let G be a plane graph of girth at least five. We show that if there exists a 3-coloring phi of a cycle C of G that does not extend to a 3-coloring of G, then G has a subgraph H on O(|C|) vertices that also has no 3-coloring extending phi. This is asymptotically best possible and improves a previous bound of Thomassen. In the next paper of the series we will use this result and the attendant theory to prove a generalization to graphs on surfaces with several precolored cycles.Comment: 48 pages, 4 figures This version: Revised according to reviewer comment

Three-coloring triangle-free graphs on surfaces I. Extending a coloring to a disk with one triangle

Let G be a plane graph with exactly one triangle T and all other cycles of length at least 5, and let C be a facial cycle of G of length at most six. We prove that a 3-coloring of C does not extend to a 3-coloring of G if and only if C has length exactly six and there is a color x such that either G has an edge joining two vertices of C colored x, or T is disjoint from C and every vertex of T is adjacent to a vertex of C colored x. This is a lemma to be used in a future paper of this series

5-list-coloring planar graphs with distant precolored vertices

We answer positively the question of Albertson asking whether every planar graph can be $5$-list-colored even if it contains precolored vertices, as long as they are sufficiently far apart from each other. In order to prove this claim, we also give bounds on the sizes of graphs critical with respect to 5-list coloring. In particular, if G is a planar graph, H is a connected subgraph of G and L is an assignment of lists of colors to the vertices of G such that |L(v)| >= 5 for every v in V(G)-V(H) and G is not L-colorable, then G contains a subgraph with O(|H|^2) vertices that is not L-colorable.Comment: 53 pages, 9 figures version 2: addresses suggestions by reviewer

Three-coloring triangle-free graphs on surfaces II. 4-critical graphs in a disk

Let G be a plane graph of girth at least five. We show that if there exists a 3-coloring of a cycle C of G that does not extend to a 3-coloring of G, then G has a subgraph H on O(|C|) vertices that also has no 3-coloring extending. This is asymptotically best possible and improves a previous bound of Thomassen. In the next paper of the series we will use this result and the attendant theory to prove a generalization to graphs on surfaces with several precolored cycles

A computational approach for finding 6-List-critical graphs on the Torus

La coloració de grafs dibuixats a superfícies és un àrea antiga i molt estudiada de la teoria de grafs. Thomassen va demostrar que hi ha un nombre finit de grafs 6-crítics a qualsevol superfície fixa i va proporcionar el conjunt explícit dels grafs 6-crítics al torus. Després, Postle va demostrar que hi ha un nombre finit de grafs 6-llista-crítics a qualsevol superfície fixa. Amb l'objectiu de trobar el conjunt de grafs 6-llista-crítics al torus, desenvolupem i implementem tècniques algorítmiques per la cerca per ordinador de grafs crítics en diferents situacions de coloració per llistes.La coloración de grafos dibujados en superficies es un área antigua y muy estudiada de la teoría de grafos. Thomassen demostró que hay un número finito de grafos 6-críticos en cualquier superficie fija y proporcionó el conjunto explícito de los grafos 6-críticos en el toro. Después, Postle demostró que hay un número finito de grafos 6-lista-críticos en cualquier superficie fija. Con el objetivo de encontrar el conjunto de grafos 6-lista-críticos en el toro, desarrollamos e implementamos técnicas algorítmicas para la búsqueda por ordenador de grafos críticos en diferentes situaciones de coloración por listas.Coloring graphs embedded on surfaces is an old and well-studied area of graph theory. Thomassen proved that there are finitely many 6-critical graphs on any fixed surface and provided the explicit set of 6-critical graphs on the torus. Later, Postle proved that there are finitely many 6-list-critical graphs on any fixed surface. With the goal of finding the set of 6-list-critical graphs on the torus, we develop and implement algorithmic techniques for computer search of critical graphs in different list-coloring settings.Outgoin