The Thue choice number versus the Thue chromatic number of graphs

Abstract

We say that a vertex colouring ϕ of a graph G is nonrepetitive if there is no positive integer n and a path on 2n vertices v1 . . . v2n in G such that the associated sequence of colours ϕ(v1) . . . ϕ(v2n) satisfy ϕ(vi) = ϕ(vi+n) for all i = 1, 2, . . . , n. The minimum number of colours in a nonrepetitive vertex colouring of G is the Thue chromatic number π(G). For the case of vertex list colourings the Thue choice number π l (G) of G denotes the smallest integer k such that for every list assignment L : V (G) → 2 N with minimum list length at least k, there is a nonrepetitive vertex colouring of G from the assigned lists. Recently it was proved that the Thue chromatic number and the Thue choice number of the same graph may have an arbitrary large difference in some classes of graphs. Here we give an overview of the known results where we compare these two parameters for several families of graphs and we also give a list of open problems on this topic.

On New Thue Colouring Concepts of Certain Graphs

“…Proof. Recall that π(P 3 ) = 2 and π(C 3 ) = 3 (see [13,15]). Hence, the result holds for non-complete graphs on 3 vertices.…”

“…From [13] it is known that π(P 1 ) = 1, π(P 2 ) = π(P 3 ) = 2 and π(P n ) = 3 for n ≥ 4. Hence, the most efficient way to reduce the Thue number of paths through edge deletion is to delete edges which create P 1 , P 2 , P 3 components.…”

“…It is known that π(K n≥1 ) = n, (see [13]). We now study the τ -index of complete graphs K n , n ≥ 3.…”

“…Certain studies on the Thue colouring of graphs have been done in [1,2,3,6,7,13,14]. Motivated by these studies, we introduce new Thue colouring concepts and study these for of certain graphs.…”

Another approach to non-repetitive colorings of graphs of bounded degree

“…One such notion that received a lot of attention is the notion of nonrepetitive coloring of graphs introduced by Currie and popularized by an article by Alon et Al. [1,6] (see [7,21] for surveys on this topic). We say that a coloring (either of the vertices or of the edges) of a graph is non-repetitive if the sequence of colors induced by any path is non-repetitive.…”