Planar graphs without cycles of length from 4 to 7 are proved to be 3-colorable. Moreover, it is proved that each proper 3-coloring of a face of length from 8 to 11 in a connected plane graph without cycles of length from 4 to 7 can be extended to a proper 3-coloring of the whole graph. This improves on the previous results on a long standing conjecture of Steinberg.

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We discuss a category of graphs, recursive clique trees, which have small-world and scale-free properties and allow a fine tuning of the clustering and the power-law exponent of their discrete degree distribution. We determine relevant characteristics of those graphs: the diameter, degree distribution, and clustering parameter. The graphs have also an interesting recursive property, and generalize recent constructions with fixed degree distributions.

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In 1982, Zaslavsky introduced the concept of a proper vertex colouring of a signed graph G as a mapping φ : V (G) → Z such that for any two adjacent vertices u and v the colour φ(u) is different from the colour σ(uv)φ(v), where is σ(uv) is the sign of the edge uv. The substantial part of Zaslavsky's research concentrated on polynomial invariants related to signed graph colourings rather than on the behaviour of colourings of individual signed graphs. We continue the study of signed graph colourings by proposing the definition of a chromatic number for signed graphs which provides a natural extension of the chromatic number of an unsigned graph. We establish the basic properties of this invariant, provide bounds in terms of the chromatic number of the underlying unsigned graph, investigate the chromatic number of signed planar graphs, and prove an extension of the celebrated Brooks theorem to signed graphs.

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To cite this version:O.V. Borodin, A.O. Ivanova, Mickael Montassier, Pascal Ochem, André Raspaud. Vertex decompositions of sparse graphs into an edgeless subgraph and a subgraph of maximum degree at most k. Journal of Graph Theory, Wiley, 2010, 65 (2) is (k, 0)-colorable. In particular, it follows that every planar graph with girth at least 7 is (8, 0)-colorable. On the other hand, we construct planar graphs with girth 6 that are not (k, 0)-colorable for arbitrarily large k.

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