Author
D. K. Skilton
Bio: D. K. Skilton is an academic researcher from IBM. The author has contributed to research in topics: Twin prime & Prime (order theory). The author has an hindex of 1, co-authored 1 publications receiving 61 citations.
Topics: Twin prime, Prime (order theory)
Papers
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TL;DR: It is proved that ifD contains {2, 3} and also contains a pair of twin primes (one of which may be 3), then four colours are necessary, and numerous results regarding periodic colourings are obtained.
Abstract: Four colours are necessary and sufficient to colour all the integers so that any two with difference equal to a prime have different colours. We investigate the corresponding problem when the setD of prescribed differences is a proper subset of the primes. In particular, we prove that ifD contains {2, 3} and also contains a pair of twin primes (one of which may be 3), then four colours are necessary. Numerous results regarding periodic colourings are also obtained. However, the problem of characterizing those setsD which necessitate four colours remains open.
62 citations
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17 Dec 1994
TL;DR: In this article, the Conjectures of Hadwiger and Hajos are used to define graph types, such as planar graph, graph on higher surfaces, and critical graph.
Abstract: Planar Graphs. Graphs on Higher Surfaces. Degrees. Critical Graphs. The Conjectures of Hadwiger and Hajos. Sparse Graphs. Perfect Graphs. Geometric and Combinatorial Graphs. Algorithms. Constructions. Edge Colorings. Orientations and Flows. Chromatic Polynomials. Hypergraphs. Infinite Chromatic Graphs. Miscellaneous Problems. Indexes.
1,380 citations
TL;DR: This paper studies the packing chromatic number of infinite distance graphs G(Z,D), i.e. graphs with the set Z of integers as vertex set, with two distinct vertices i,j@?Z being adjacent if and only if |i-j|@?D.
50 citations
TL;DR: In this paper, the chromatic number χ(Z,D) of a distance graph with distance set Z is shown to be Ω(Z + D) when |D| is finite.
Abstract: Suppose D is a subset of all positive integers. The distance graph G(Z,D) with distance set D is the graph with vertex set Z, and two vertices x and y are adjacent if and only if |x− y| ∈ D. This paper studies the chromatic number χ(Z,D) of G(Z,D). In particular, we prove that χ(Z,D) ≤ |D|+1 when |D| is finite. Exact values of χ(G,D) are also determined for some D with |D| = 3. c © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 287–294, 1997
47 citations
TL;DR: It is proved that for any finite integral setT that contains 0, the asymptoticT-coloring ratioR(T) is equal to the fractional chromatic number of the distance graphG(Z,D), whereD=T?{0}.
45 citations
01 Jan 2007
TL;DR: A List of Symbols Used x as discussed by the authors is a list of symbols used in the past and present of symbols in the English language, including symbols used for symbol-based communication.
Abstract: ix List of Symbols Used x
44 citations