Mariusz Hałuszczak

Nonrepetitive colorings of graphs

A sequence a = a1a2. . . . an is said to be nonrepetitive if no two adjacent blocks of a are exactly the same. For instance, the sequence 1232321 contains a repetition 2323, while 123132123213 is nonrepetitive. A theorem of Thue asserts that, using only three symbols, one can produce arbitrarily long nonrepetitive sequences. In this paper we consider a natural generalization of Thues sequences for colorings of graphs. A coloring of the set of edges of a given graph G is nonrepetitive if the sequence of colors on any path in G is nonrepetitive. 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 Δ(G). We show, by an application of the Lovasz Local Lemma, that the Thue number stays bounded for graphs with bounded maximum degree, in particular, π(G) ≤ cΔ(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(Δ(T - 1) for any tree T with at least two edges. We conclude by discussing some generalizations and proposing several problems and conjectures.

Schütte's Tournament Problem and Intersecting Families of Sets

In this paper we consider a bipartite version of Schuttes well-known tournament problem. A bipartite tournament

On the structure and deficiency of k -trees with bounded degree

T=(A,B,E)

On Ramsey ( K 1, 2 , K n )-minimal graphs.

with teams

Acyclic reducible bounds for outerplanar graphs

A

About acyclic edge colourings of planar graphs

and

On-line Ramsey Theory

B

On Ramsey minimal graphs

, and set of arcs

On Ramsey (K1,m,G)-minimal graphs

E

Acyclic chromatic indices of fully subdivided graphs

, has the property