Graph Colouring and Decomposition

Abstract

A (vertex) ℓ-ranking is a colouring φ : V (G) → N of the vertices of a graph G with integer colours so that for any path u 0 , . . . , u p of length at most ℓ, φ(u 0 ) ̸ = φ(u p ) or φ(u 0 ) < max{φ(u 0 ), . . . , φ(u p )}. We show that, for any őxed integer ℓ ≥ 2, every n-vertex planar graph has an ℓ-ranking using O(log n/ log log log n) colours and this is tight even when ℓ = 2; for inőnitely many values of n, there are n-vertex planar graphs, for which any 2-ranking requires Ω(log n/ log log log n) colours. This result also extends to bounded genus graphs.In developing this proof we obtain optimal bounds on the number of colours needed for ℓ-ranking graphs of treewidth t and graphs of simple treewidth t. These upper bounds are constructive and give O(n)-time algorithms. Additional results that come from our techniques include new sublogarithmic upper bounds on the number of colours needed for ℓ-rankings of apex minor-free graphs and k-planar graphs.