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.

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Graph Coloring Problems

On packing colorings of distance graphs

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

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Integral distance graphs

https://ir.nctu.edu.tw:443/bitstream/11536/31460/1/000079054600007.pdf

Distance Graphs andT-Coloring

/pdf/independence-polynomials-of-circulant-graphs-2ydibivtem.pdf

Independence polynomials of circulant graphs

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.

D. K. Skilton

Papers

Colouring prime distance graphs