Abstract :
In this paper we consider colorings of graphs avoiding certain patterns on paths. Let image be a set of variables and let image, be a pattern, that is, any sequence of variables. A finite sequence image is said to match a pattern image if image may be divided into non-empty blocks image, such that image implies image, for all image. A coloring of vertices (or edges) of a graph image is said to be image-free if no path in image matches a pattern image. The pattern chromatic number image is the minimum number of colors used in a image-free coloring of image.
Extending the result of Alon et al. [Non-repetitive colorings of graphs, Random Struct. Alg. 21 (2002) 336–346] we prove that if each variable occurs in a pattern image at least image times then image, where image is an absolute constant. The proof is probabilistic and uses the Lovász Local Lemma. We also provide some explicit image-free colorings giving stronger estimates for some simple classes of graphs. In particular, for some patterns image we show that image is absolutely bounded by a constant depending only on image, for all trees image.
