The game colouring number of powers of forests

TL;DR

This paper establishes a new upper bound for the game colouring number of powers of forests, significantly improving previous bounds and contributing to graph theory and combinatorics.

Contribution

It provides a tighter upper bound for the game colouring number of powers of forests, advancing understanding in graph coloring theory.

Findings

New upper bound for game colouring number of powers of forests

Improvement over previous bounds by an asymptotic factor of 2

Applicable for forests with maximum degree at least 3

Abstract

We prove that the game colouring number of the $m$-th power of a forest of maximum degree $\Delta\ge3$ is bounded from above by \[\frac{(\Delta-1)^m-1}{\Delta-2}+2^m+1,\] which improves the best known bound by an asymptotic factor of 2.