Ordered colourings Original Research Article

Let k be a positive integer. An ordered k-colouring of a graph G is a function c from V(G) into {1,…,k} such that for every pair of distinct vertices x and y and for every (x, y)-path P, if c(x) = c(y), then there exists an internal vertex z of P such that c(x) < c(z). We prove some theorems on ordered colourings of trees and planar graphs, and examine the relationship between connectivity and ordered colourings. Let A be a set of graphs which is ordered by subgraphs and closed under subgraphs. We characterize when A is a well-quasi-order. A generalization of ordered colourings is given.