The elements of a plane graph G are the edges, vertices, and faces of G. We say that two elements are neighbours in G if they are incident with or are mutually adjacent with each other in G. The simultaneous colouring of distinct elements of a planar graph was first introduced by Ringel [12]. In his paper Ringel considered the problem of colouring the vertices and faces of a plane graph in such...

A p–colouring of a knot K is a surjective homomorphism ρ from its knot group G := π1(S − K) to D2p := {t, s|t2 = sp = 1, tst = sp−1} the dihedral group of order 2p, when p is any odd integer. The pair (K, ρ) is called a p–coloured knot. It is a well-known fact that we can encode ρ as a colouring of arcs of a knot diagram by elements of Zp (the cyclic group of order p), subject to the ‘colouring...

In an edge-coloured host graph G, a subgraph H is properly coloured if no two incident edges of H receive the same colour, and rainbow if no two edges of H receive the same colour. Given a positive integer k, a host graph G, an edge-colouring c of G (c is not necessarily proper), then c is a k-colouring if c uses k colours overall, c is a local k-colouring if at most k colours are used at each ...

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) is denoted by dG(x), the maximum degree of a vertex of G-by ∆(G), and the chromatic index of G-by χ(G) a...

Here we will discuss a winning strategy of impossible colouring pseudo-telepathy game for the set of vectors having KochenSpecker property in four dimension with single use of NLbox. Then we discuss some sufficient condition for the winning strategy of impossible colouring pseudo-telepathy game for general d-dimension with single use of NL-box.

An edge colouring of a graph G is called acyclic if it is proper and every cycle contains at least three colours. We show that for every ε > 0, there exists a g = g(ε) such that if G has girth at least g then G admits an acyclic edge colouring with at most (1 + ε)∆ colours.

We study the complexity of the colouring problem for circle graphs. We will solve the two open questions of [Un88], where first results were presented. 1. Here we will present an algorithm which solves the 3-colouring problem of circle graphs in time O(n log(n)). In [Un88] we showed that the 4-colouring problem for circle graphs is NP-complete. 2. If the largest clique of a circle graph has siz...

An instance of List Colouring consists of a graph G and a list L(v) of colours for each vertex v of G. We are asked to determine if there is an acceptable colouring of G, that is a colouring in which each vertex receives a colour from its list, and no edge has both its endpoints coloured with the same colour. The list-chromatic number of G, denoted χl(G) is the minimum integer k such that for e...

Biclique-colouring is a colouring of the vertices of a graph in such a way that no maximal complete bipartite subgraph with at least one edge is monochromatic. We show that it is coNP-complete to check whether a given function that associates a colour to each vertex is a bicliquecolouring, a result that justifies the search for structured classes where the biclique-colouring problem could be ef...

This paper studies on-line list colouring of graphs. It is proved that the online choice number of a graph G on n vertices is at most χ(G) ln n + 1, and the on-line b-choice number of G is at most eχ(G)−1 e−1 (b − 1 + lnn) + b. Suppose G is a graph with a given χ(G)-colouring of G. Then for any (χ(G) ln n + 1)-assignment L of G, we give a polynomial time algorithm which constructs an L-colourin...