We show that the vertices of any plane graph in which every face is of length at least g can be colored by ⌊(3g − 5)/4⌋ colors so that every color appears in every face. This is nearly tight, as there are plane graphs that admit no vertex coloring of this type with more than ⌊(3g + 1)/4⌋ colors. We further show that the problem of determining whether a plane graph admits a vertex coloring by 3 colors in which all colors appear in every face is N P-complete even for graphs in which all faces are of length 3 or 4 only. If all faces are of length 3 this can be decided in polynomial time. The investigation of this problem is motivated by its connection to a variant of the art gallery problem in computational geometry.

We show that any graph of maximum degree at most 3 has a two-coloring such that one color-class is an independent set, while the other color-class induces monochromatic components of order at most 750. On the other hand, for any constant C, we exhibit a 4-regular graph such that the deletion of any independent set leaves at least one component of order greater than C. Similar results are obtained for coloring graphs of given maximum degree with k + colors such that k parts form an independent set and parts span components of order bounded by a constant. A lot of interesting questions remain open.

We study the concept of propagation connectivity on random 3-uniform hypergraphs. This concept is inspired by a simple linear time algorithm for solving instances of certain constraint satisfaction problems. We derive upper and lower bounds for the propagation connectivity threshold, and point out some algorithmic implications.1 A part of this work has been done while the first and the third authors were visiting

We study relaxations of proper two-colourings, such that the order of the induced monochromatic components in one (or both) of the colour classes is bounded by a constant. A colouring of a graph G is called (C 1 , C 2 )-relaxed if every monochromatic component induced by vertices of the first (second) colour is of order at most C 1 (C 2 , resp.). We prove that the decision problem 'Is there a (1, C)-relaxed colouring of a given graph G of maximum degree 3?' exhibits a hardness jump in the component order C. In other words, there exists an integer f(3) such that the decision problem is NP-hard for every 2 C < f(3), while every graph of maximum degree 3 is (1, f(3))-relaxed colourable. We also show f(3) 22 by way of a quasilinear time algorithm, which finds a (1, 22)-relaxed colouring of any graph of maximum degree 3. Both the bound on f(3) and the running time greatly improve earlier results. We also study the symmetric version, that is, when C 1 = C 2 , of the relaxed colouring problem and make the first steps towards establishing a similar hardness jump.

We show that the vertices of any plane graph in which every face is incident to at least g vertices can be colored by (3g − 5)/4 colors so that every color appears in every face. This is nearly tight, as there are plane graphs where all faces are incident to at least g vertices and that admit no vertex coloring of this type with more than (3g + 1)/4 colors. We further show that the problem of determining whether a plane graph admits a vertex coloring by k colors in which all colors appear in every face is in P for k = 2 and is N P-complete for k = 3, 4. We refine this result for polychromatic 3-colorings restricted to 2-connected graphs which have face sizes from a prescribed (possibly infinite) set of integers. Thereby we find an almost complete characterization of these sets of integers (face sizes) for which the corresponding decision problem is in P, and for the others it is N P-complete.