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

TL;DR

This paper introduces a new, more elementary proof technique for non-repetitive graph colorings that matches entropy-compression bounds and improves existing upper bounds for graphs with bounded degree.

Contribution

The authors present a novel proof method applicable to non-repetitive colorings, providing shorter, more elementary proofs and improved upper bounds for graphs of bounded degree.

Findings

Improved upper bound on non-repetitive number: 4.25Δ + o(Δ)

New upper bound on weak total non-repetitive number: 4.25Δ + o(Δ)

Enhanced bound on total non-repetitive number: Δ^2 + (3/2^{1/3})Δ^{5/3} + o(Δ^{5/3})

Abstract

We propose a new proof technique that aims to be applied to the same problems as the Lov\'asz 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 numbers to the maximal degree of a graph. It seems that there should be other interesting applications to the presented approach. In terms of upper-bound our approach seems to be as strong as entropy-compression, but the proofs are more elementary and shorter. The application 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 number, a $4.25\Delta +o(\Delta)$ upper-bound on the weak total non-repetitive number and a $ \Delta^2+\frac{3}{2^\frac{1}{3}}\Delta^{\frac{5}{3}}+ o(\Delta^{\frac{5}{3}})$ upper-bound on the total non-repetitive number of graphs. This last result implies the same upper-bound for the non-repetitive index of graphs, which improves the best known bound.