Another approach to non-repetitive colorings of graphs of bounded degree (Q2195220)

Summary: We propose a new proof technique that applies to the same problems as the Lovász Local Lemma or the entropy-compression method. We present this approach in the context of non-repetitive colorings and we use it to improve upper-bounds relating different non-repetitive chromatic numbers to the maximal degree of a graph. It seems that there should be other interesting applications of the presented approach. In terms of upper-bounds our approach seems to be as strong as entropy-compression, but the proofs are more elementary and shorter. The applications we provide in this paper are upper bounds for graphs of maximal degree at most \(\Delta \): a minor improvement on the upper-bound of the non-repetitive chromatic number, a \(4.25\Delta +o(\Delta)\) upper-bound on the weak total non-repetitive chromatic number, and a \(\Delta^2+\frac{3}{2^{1/3}}\Delta^{5/3}+ o(\Delta^{5/3})\) upper-bound on the total non-repetitive chromatic number of graphs. This last result implies the same upper-bound for the non-repetitive chromatic index of graphs, which improves the best known bound.