Abstract. A sequence a = a 1 a 2 ...an is said to be non-repetitive if no two adjacent blocks of a are exactly the same. For instance the sequence 1232321 contains a repetition 2323, while 123132123213 is non-repetitive. A theorem of Thue asserts that, using only three symbols, one can produce arbitrarily long non-repetitive sequences. In this paper we consider a natural generalization of Thue's sequences for colorings of graphs. A coloring of the set of edges of a given graph G is non-repetitive if the sequence of colors on any path in G is non-repetitive. We call the minimal number of colors needed for such a coloring the Thue number of G and denote it by π(G).The main problem we consider is the relation between the numbers π(G) and ∆(G). We show, by an application of the Lovász Local Lemma, that the Thue number stays bounded for graphs with bounded maximum degree, in particular, π(G) ≤ c∆(G) 2 for some absolute constant c. For certain special classes of graphs we obtain linear upper bounds on π(G), by giving explicit colorings. For instance, the Thue number of the complete graph Kn is at most 2n − 3, and π(T ) ≤ 4(∆(T ) − 1) for any tree T with at least two edges. We conclude by discussing some generalizations and proposing several problems and conjectures.
The Ramsey game we consider in this paper is played on an unbounded set of vertices by two players, called Builder and Painter. In one move Builder introduces a new edge and Painter paints it red or blue. The goal of Builder is to force Painter to create a monochromatic copy of a fixed target graph $H$, keeping the constructed graph in a prescribed class ${\cal G}$. The main problem is to recognize the winner for a given pair $H,{\cal G}$. In particular, we prove that Builder has a winning strategy for any $k$-colorable graph $H$ in the game played on $k$-colorable graphs. Another class of graphs with this strange self-unavoidability property is the class of forests. We show that the class of outerplanar graphs does not have this property. The question of whether planar graphs are self-unavoidable is left open. We also consider a multicolor version of Ramsey on-line game. To extend our main result for $3$-colorable graphs we introduce another Ramsey type game, which seems interesting in its own right.
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