A sequence of the form s 1 s 2 . . . s m s 1 s 2 . . . s m is called a repetition. A vertex-coloring of a graph is called nonrepetitive if none of its paths is repetitively colored. We answer a question of Grytczuk [Thue type problems for graphs, points and numbers, manuscript] by proving that every outerplanar graph has a nonrepetitive 12-coloring. We also show that graphs of tree-width t have nonrepetitive 4 t -colorings.

An edge in a drawing of a graph is called even if it intersects every other edge of the graph an even number of times. Pach and Tóth proved that a graph can always be redrawn so that its even edges are not involved in any intersections. We give a new and significantly simpler proof of the stronger statement that the redrawing can be done in such a way that no new odd intersections are introduced. We include two applications of this strengthened result: an easy proof of a theorem of Hanani and Tutte (the only proof we know of not to use Kuratowski's theorem), and the new result that the odd crossing number of a graph equals the crossing number of the graph for values of at most 3. The paper begins with a disarmingly simple proof of a weak (but standard) version of the theorem by Hanani and Tutte.

Given lists of available colors assigned to the vertices of a graph G, a list coloring is a proper coloring of G such that the color on each vertex is chosen from its list. If the lists all have size k, then a list coloring is equitable if each color appears on at most dn(G )/ke vertices. A graph is equitably k-choosable if such a coloring exists whenever the lists all have size k. We prove that G is equitably k-choosable when k ! maxfÁ(G ),n(G )/ ------------------ Kostochka and West

A drawing of a graph is x-monotone if every edge intersects every vertical line at most once and every vertical line contains at most one vertex. Pach and Tóth showed that if a graph has an x-monotone drawing in which every pair of edges crosses an even number of times, then the graph has an x-monotone embedding in which the x-coordinates of all vertices are unchanged. We give a new proof of this result and strengthen it by showing that the conclusion remains true even if adjacent edges are allowed to cross each other oddly. This answers a question posed by Pach and Tóth. We show that a further strengthening to a "removing even crossings" lemma is impossible by separating monotone versions of the crossing and the odd crossing number.Our results extend to level-planarity, which is a well-studied generalization of x-monotonicity. We obtain a new and simple algorithm to test level-planarity in quadratic time, and we show that x-monotonicity of edges in the definition of level-planarity can be relaxed.

In 1996, Matheson and Tarjan conjectured that any n-vertex plane triangulation with n sufficiently large has a dominating set of size at most n/4. We prove this for graphs of maximum degree 6.