Abstract
Let D be a set of positive integers. The kappa value of D, denoted by \(\kappa (D)\), is the parameter involved in the so called “lonely runner conjecture.” Let x, y be positive integers, we investigate the kappa values for the family of sets \(D =\{2, 3, x, y\}\). For a fixed positive integer \(x > 3\), the exact values of \(\kappa (D)\) are determined for \(y=x+i\), \(1 \le i \le 6\). These results lead to some asymptotic behavior of \(\kappa (D)\) for \(D=\{2, 3, x, y\}\).
Supported in part by the National Science Foundation under grant DMS-1247679.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Barajas, J., Serra, O.: The lonely runner with seven runners. Electron. J. Combin. 15, #R48 (2008)
Bienia, W., Goddyn, L., Gvozdjak, P., Sebő, A., Tarsi, M.: Flows, view obstructions, and the lonely runner. J. Combin. Theory Ser. B 72, 1–9 (1998)
Bohman, T., Holzman, R., Kleitman, D.: Six lonely runners. Electron. J. Comb. 8, 49 (2001). Research Paper 3
Cantor, D., Gordon, B.: Sequences of integers with missing differences. J. Comb. Theory Ser. A 14, 281–287 (1973)
Chang, G., Liu, D., Zhu, X.: Distance graphs and \(T\)-coloring. J. Comb. Theory Ser. B 75, 159–169 (1999)
Cusick, T.: View-obstruction problems in \(n\)-dimensional geometry. J. Comb. Theory Ser. A 16, 1–11 (1974)
Cusick, T., Pomerance, C.: View-obstruction problems, III. J. Number Theory 19, 131–139 (1984)
Eggleton, R., Erdős, P., Skilton, D.: Colouring the real line. J. Comb. Theory Ser. A 39, 86–100 (1985)
Eggleton, R., Erdős, P., Skilton, D.: Research problem 77. Discrete Math. 58, 323 (1986)
Eggleton, R., Erdős, P., Skilton, D.: Colouring prime distance graphs. Graphs Comb. 6, 17–32 (1990)
Gupta, S.: Sets of integers with missing differences. J. Comb. Theory Ser. A 89, 55–69 (2000)
Haralambis, N.: Sets of integers with missing differences. J. Comb. Theory Ser. A 23, 22–33 (1997)
Kemnitz, A., Kolberg, H.: Coloring of integer distance graphs. Discrete Math. 191, 113–123 (1998)
Liu, D.: From rainbow to the lonely runner: a survey on coloring parameters of distance graphs. Taiwanese J. Math. 12, 851–871 (2008)
Liu, D., Sutedja, A.: Chromatic number of distance graphs generated by the sets \(\{2, 3, x, y\}\). J. Comb. Optim. 25, 680–693 (2013)
Liu, D., Zhu, X.: Fractional chromatic number and circular chromatic number for distance graphs with large clique size. J. Graph Theory 47, 129–146 (2004)
Rabinowitz, J., Proulx, V.: An asymptotic approach to the channel assignment problem. SIAM J. Alg. Discrete Methods 6, 507–518 (1985)
Voigt, M., Walther, H.: Chromatic number of prime distance graphs. Discrete Appl. Math. 51, 197–209 (1994)
Voigt, M., Walther, H.: On the chromatic number of special distance graphs. Discrete Math. 97, 395–397 (1991)
Wills, J.: Zwei Sätze über inhomogene diophantische appromixation von irrationlzahlen. Monatsch. Math. 71, 263–269 (1967)
Zhu, X.: Circular chromatic number: a survey. Discrete Math. 229, 371–410 (2001)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Collister, D., Liu, D.DF. (2015). Study of \(\kappa (D)\) for \(D = \{2, 3, x, y\}\) . In: Jan, K., Miller, M., Froncek, D. (eds) Combinatorial Algorithms. IWOCA 2014. Lecture Notes in Computer Science(), vol 8986. Springer, Cham. https://doi.org/10.1007/978-3-319-19315-1_22
Download citation
DOI: https://doi.org/10.1007/978-3-319-19315-1_22
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-19314-4
Online ISBN: 978-3-319-19315-1
eBook Packages: Computer ScienceComputer Science (R0)