Boštjan Brešar

Nonrepetitive colorings of trees

A coloring of the vertices of a graph G is nonrepetitive if no path in G forms a sequence consisting of two identical blocks. The minimum number of colors needed is the Thue chromatic number, denoted by @p(G). A famous theorem of Thue asserts that @p(P)=3 for any path P with at least four vertices. In this paper we study the Thue chromatic number of trees. In view of the fact that @p(T) is bounded by 4 in this class we aim to describe the 4-chromatic trees. In particular, we study the 4-critical trees which are minimal with respect to this property. Though there are many trees T with @p(T)=4 we show that any of them has a sufficiently large subdivision H such that @p(H)=3. The proof relies on Thue sequences with additional properties involving palindromic words. We also investigate nonrepetitive edge colorings of trees. By a similar argument we prove that any tree has a subdivision which can be edge-colored by at most @D+1 colors without repetitions on paths.

Hypercubes As Direct Products

Let G be a connected bipartite graph. An involution

On the geodetic number and related metric sets in Cartesian product graphs

\alpha

On Vizing's conjecture

of G that preserves the bipartition of G is called bipartite. Let

Partial hamming graphs and expansion procedures

G^\alpha

A generalization of Hungarian method and Hall's theorem with applications in wireless sensor networks

be the graph obtained from G by adding to G the natural perfect matching induced by

Paired-domination of Cartesian products of graphs and rainbow domination

\alpha

On the Natural Imprint Function of a Graph

. We show that the k-cube Qk is isomorphic to the direct product

Cover-Incomparability Graphs of Posets

G \times H

On the weighted k-path vertex cover problem

if and only if G is isomorphic to