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Colouring random geometric graphs

by C. J. H. Mcdiarmid, T. Müller
"... 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 functio ..."
Abstract - Cited by 1 (1 self) - Add to MetaCart
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

5-colouring graphs with 4 crossings ∗

by Rok Erman A, Frédéric Havet B, Bernard Lidicky ́ C , 2010
"... We disprove a conjecture of Oporowski and Zhao stating that every graph with crossing number at most 5 and clique number at most 5 is 5-colourable. However, we show that every graph with crossing number at most 4 and clique number at most 5 is 5-colourable. We also show some colourability results on ..."
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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

Edge Colouring Reduced Indifference Graphs

by Celina M. H. De Figueiredo, Joao Meidanis, Célia Picinin de Mello, Carmen Ortiz , 1999
"... 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 ..."
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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

Acyclic Colourings of 1-Planar Graphs

by O.V. Borodin, A. V. Kostochka, A. Raspaud, E. Sopena , 2001
"... 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. ..."
Abstract - Cited by 11 (0 self) - Add to MetaCart
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.

Interval Edge Colourings of Complete Graphs and n-cubes

by Petros A. Petrosyan , 2007
"... 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 den ..."
Abstract - Cited by 1 (1 self) - Add to MetaCart
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

Graph homomorphisms, circular colouring, and fractional covering by hcuts

by Robert Engström, Tommy Färnqvist, Peter Jonsson
"... 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 t ..."
Abstract - Cited by 4 (3 self) - Add to MetaCart
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

List Edge Colourings of Some 1-Factorable Multigraphs

by M. N. Ellingham, Luis Goddyn , 1996
"... 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 ..."
Abstract - Cited by 25 (0 self) - Add to MetaCart
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

Bipartite edge-colouring in O(∆m) time

by Alexander Schrijver , 1996
"... 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 minimu ..."
Abstract - Cited by 7 (1 self) - Add to MetaCart
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

A computational analysis of colour constancy

by Donald I A Macleod , Jürgen Golz - Oxford University , 2003
"... 2 The visual stimulus associated with an object surface varies with the illumination falling on the object. Constancy of perceived surface colour under changing illumination implies that the illumination-dependent mapping from surface reflectance to stimulus is cancelled by a compensatory variation ..."
Abstract - Cited by 7 (0 self) - Add to MetaCart
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

COLOURING LATTICE POINTS BY REAL NUMBERS

by D. G. Fon-der-flaass , 2005
"... 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 ..."
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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
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