 
Summary: Breaking the rhythm on graphs
Noga Alon and Jaroslaw Grytczuk
Abstract. We study graph colorings avoiding periodic sequences with
large number of blocks on paths. The main problem is to decide, for
a given class of graphs F, if there are absolute constants t, k such that
any graph from the class has a tcoloring with no k identical blocks in
a row appearing on a path. The minimum t for which there is some k
with this property is called the rhythm threshold of F, denoted by t(F).
For instance, we show that the rhythm threshold of graphs of maximum
degree at most d is between (d+1)/2 and d+1. We give several general
conditions for finiteness of t(F), as well as some connections to existing
chromatic parameters. The question whether the rhythm threshold is
finite for planar graphs remains open.
1. Introduction
Let k 2 be a fixed integer. A vertex coloring f of a graph G is k
repetitive if there is a positive integer n and a path on kn vertices v1, v2, ..., vkn
such that f(vi) = f(vi+n) = . . . = f(vi+(k1)n) for all 1 i n. That is,
if there is at least one path in G that looks like a periodic sequence with
k blocks. Otherwise f is called knonrepetitive. In this case there are no k
identical blocks in a row on any path of G. This type of coloring is a graph
