We present results on partitioning the vertices of $2$-edge-colored graphs into monochromatic paths and cycles. We prove asymptotically the two-color case of a conjecture of S\'ark\"ozy: the vertex set of every $2$-edge-colored graph can be partitioned into at most $2\alpha(G)$ monochromatic cycles, where $\alpha(G)$ denotes the independence number of $G$. Another direction, emerged recently from a conjecture of Schelp, is to consider colorings of graphs with given minimum degree. We prove that apart from $o(|V(G)|)$ vertices, the vertex set of any $2$-edge-colored graph $G$ with minimum degree at least $(1+\eps){3|V(G)|\over 4}$ can be covered by the vertices of two vertex disjoint monochromatic cycles of distinct colors. Finally, under the assumption that $\overline{G}$ does not contain a fixed bipartite graph $H$, we show that in every $2$-edge-coloring of $G$, $|V(G)|-c(H)$ vertices can be covered by two vertex disjoint paths of different colors, where $c(H)$ is a constant depending only on $H$. In particular, we prove that $c(C_4)=1$, which is best possible

Abstract. Directed path-width was defined by Reed, Thomas and Seymour around 1995. The author and P. Hajnal defined a cops-and-robber game on digraphs in 2000. We prove that the two notions are closely related and for any digraph D, the corresponding graph parameters differ by at most one. The result is achieved using the mixed-search technique developed by Bienstock and Seymour. A search is called monotone, in which the robber's territory never increases. We show that there is a mixed-search of D with k cops if and only if there is a monotone mixed-search with k cops. For our cops-and-robber game we get a slightly weaker result: the monotonicity can be guaranteed by using at most one extra cop.

A sequence of symbols a1, a2 . . . is called square-free if it does not contain a subsequence of consecutive terms of the form x1, . . . , xm, x1, . . . , xm. A century ago Thue showed that there exist arbitrarily long square-free sequences using only three symbols. Sequences can be thought of as colors on the vertices of a path. Following the paper of Alon, Grytczuk, Ha luszczak and Riordan, we examine graph colorings for which the color sequence is square-free on any path. The main result is that the vertices of any k-tree have a coloring of this kind using O(c k ) colors if c > 6. Alon et al. conjectured that a fixed number of colors suffices for any planar graph. We support this conjecture by showing that this number is at most 12 for outerplanar graphs. On the other hand we prove that some outerplanar graphs require at least 7 colors. Using this latter we construct planar graphs, for which at least 10 colors are necessary.

A vertex colouring of a graph is nonrepetitive on paths if there is no path v1, v2, . . . , v2t such that vi and vt+i receive the same colour for all i = 1, 2, . . . , t. We determine the maximum density of a graph that admits a k-colouring that is nonrepetitive on paths. We prove that every graph has a subdivision that admits a 4-colouring that is nonrepetitive on paths. The best previous bound was 5. We also study colourings that are nonrepetitive on walks, and provide a conjecture that would imply that every graph with maximum degree ∆ has a f (∆)-colouring that is nonrepetitive on walks. We prove that every graph with treewidth k and maximum degree ∆ has a O(k∆)-colouring that is nonrepetitive on paths, and a O(k∆ 3 )-colouring that is nonrepetitive on walks.

We conjecture that, for each tree T , there exists a natural number k T such that the following holds: If G is a k T -edge-connected graph such that |E(T )| divides |E(G)|, then the edges of G can be divided into parts, each of which is isomorphic to T . We prove that for T = K 1,3 (the claw), this holds if and only if there exists a (smallest) natural number k t such that every k t -edge-connected graph has an orientation for which the indegree of each vertex equals its outdegree modulo 3. Tutte's 3-flow conjecture says that k t = 4. We prove the weaker statement that every 4 log n -edge-connected graph with n vertices has an edge-decomposition into claws provided its number of edges is divisible by 3. We also prove that every triangulation of a surface has an edge-decomposition into claws.