On Distance Coloring

Call a connected undirected graph (d,c)-colorable if there is a vertex coloring using at most c colors such that no two vertices of distance d or less have the same color. It is well known that (1,2)-colorability is decidable in linear time, but (1,c)-colorability for c greater than or equal to 3 is NP-complete. Sharp (2007) shows that for fixed d greater than or equal to 2, the (d,c)-colorability problem is solvable in linear time for c less than or equal to 3d/2 and NP-complete otherwise. In this note we give an alternative construction that improves the upper time bound as a function of d for the case c less than or equal to 3d/2. The construction entails a generalization of the notion of tree decomposition and bounded treewidth (Robertson and Seymour 1986) to arbitrary overlay graphs, not just trees, which may be of independent interest

Embedded graph 3-coloring and flows

A graph drawn in a surface is a near-quadrangulation if the sum of the lengths of the faces different from 4-faces is bounded by a fixed constant. We leverage duality between colorings and flows to design an efficient algorithm for 3-precoloring-extension in near-quadrangulations of orientable surfaces. Furthermore, we use this duality to strengthen previously known sufficient conditions for 3-colorability of triangle-free graphs drawn in orientable surfaces.Comment: 53 pages, 15 figure

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

Algorithmic Graph Theory

The main focus of this workshop was on mathematical techniques needed for the development of efficient solutions and algorithms for computationally difficult graph problems. The techniques studied at the workshhop included: the probabilistic method and randomized algorithms, approximation and optimization, structured families of graphs and approximation algorithms for large problems. The workshop Algorithmic Graph Theory was attended by 46 participants, many of them being young researchers. In 15 survey talks an overview of recent developments in Algorithmic Graph Theory was given. These talks were supplemented by 10 shorter talks and by two special sessions