Chromatic number of distance graphs generated by the sets {2,3,x,y}

Daphne Der-Fen Liu () and Aileen Sutedja ()
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Daphne Der-Fen Liu: California State University
Aileen Sutedja: California State University

Journal of Combinatorial Optimization, 2013, vol. 25, issue 4, No 15, 680-693

Abstract: Abstract Let D be a set of positive integers. The distance graph generated by D has all integers ℤ as the vertex set; two vertices are adjacent whenever their absolute difference falls in D. We completely determine the chromatic number for the distance graphs generated by the sets D={2,3,x,y} for all values x and y. The methods we use include the density of sequences with missing differences and the parameter involved in the so called “lonely runner conjecture”. Previous results on this problem include: For x and y being prime numbers, this problem was completely solved by Voigt and Walther (Discrete Appl. Math. 51:197–209, 1994); and other results for special integers of x and y were obtained by Kemnitz and Kolberg (Discrete Math. 191:113–123, 1998) and by Voigt and Walther (Discrete Math. 97:395–397, 1991).

Keywords: Distance graphs; Chromatic number; Density of sequences with missing differences; Lonely runner conjecture (search for similar items in EconPapers)
Date: 2013
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10878-012-9509-4

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