some probability distribution ν on R d). For i � = j we join Xi and Xj by an edge if �Xi − Xj �< r(n). We study the properties of the chromatic number χn and clique number ωn of this graph as n becomes large, where we assume that r(n) → 0. We allow any choice ν that has a bounded density

on graphs that can be made planar by removing few edges. In particular, we show that if a graph with clique number at most 5 has three edges whose removal leaves the graph planar, then it is 5-colourable. 1

The chromatic index problem -- finding the minimum number of colours required for colouring the edges of a graph -- is still unsolved for indifference graphs, whose vertices can be linearly ordered so that the vertices contained in the same maximal clique are consecutive in this order. Two adjacent

A graph is 1-planar if it can be drawn on the plane in such a way that every edge crosses at most one other edge. We prove that the acyclic chromatic number of every 1-planar graph is at most 20.

For complete graphs and n-cubes bounds are found for the possible number of colours in an interval edge colourings. Let G = (V, E) be an undirected graph without loops and multiple edges [1], V (G) and E(G) be the sets of vertices and edges of G, respectively. The degree of a vertex x ∈ V (G

A graph homomorphism is a vertex map which carries edges from a source graph to edges in a target graph. The instances of the Weighted Maximum H-Colourable Subgraph problem (MAX H-COL) are edge-weighted graphs G and the objective is to find a subgraph of G that has maximal total edge weight, under

The List Edge Colouring Conjecture asserts that, given any multigraph G with chromatic index k and any set system fSe : e 2 E(G)g with each jSe j = k, we can choose elements se 2 Se such that se 6= sf whenever e and f are adjacent edges. Using a technique of Alon and Tarsi which involves the graph

We show that a minimum edge-colouring of a bipartite graph can be found in O(∆m) time, where ∆ and m denote the maximum degree and the number of edges of G, respectively. It is equivalent to finding a perfect matching in a k-regular bipartite graph in O(km) time. By sharpening the methods, a

retinal stimulus to perceived colour. Can this be done-and if so, what is the nature of the required compensatory mapping? We investigate this question for four quite different idealizations of the visual environment, discussing these in turn in the context of a very simple proposal about the nature

Abstract. We establish a criterion for the existence of an f-colouring with a finite span of the d-dimensional lattice graph Zd. Let G be an arbitrary connected simple graph. By d(u, v) we denote the graph distance between the vertices u, v of G. By a constraints function we mean any non-increasing