Abstract
The following generalization of graph colouring arises naturally in the study of channel assignment for cellular radiocommunications networks.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
S.M. Allen and N. Dunkin, Frequency assignment problems: representations and solutions, Technical report, University of Glamorgan, 1997.
S.M. Allen, D.H. Smith and S. Hurley, Lower bounding techniques for frequency assignment, Discrete Mathematics 197/198 (1999) 41–52.
G. Ausiello, P. Crescenzi, G. Gambosi, V. Kann, A. Marchetti-Spaccamela and M. Protasi, Complexity and Approximation, Springer, 1999.
F. Barasi and J. van den Heuvel, Graph labelling, acyclic orientations, and greedy algorithms, manuscript, 2001.
M. Bernstein, N.J.A. Sloan and P.E. Wright, On sublattices of the hexagonal lattice, Discrete Mathematics 170 (1997) 29–39.
B. Bollobás, The chromatic number of random graphs, Combinatorica 8 (1988) 49–55.
B. Bollobas, Modem Graph Theory, Graduate Texts in Mathematics 184, Springer, 1998.
H. Breu and D.G. Kirkpatrick, Unit disk graph recognition is NP-hard, Comput. Geom. 9 (1998) 3–24.
S. Ceroi, Clique number of intersection graphs of convex bodies, manuscript, 2001.
G.J. Chang and D. Kuo, The L(2, l)-labeling problem on graphs, SIAM J. Discrete Math. 9 (1996) 309–316.
G. Chang, L. Huang and X. Zhu, Circular chromatic numbers and fractional chromatic numbers of distance graphs, Discrete Math. 19 (1997) 223–230.
B.N. Clark, C.J. Colbourn and D.S. Johnson, Unit disk graphs, Discrete Mathematics 86 (1990) 165–177.
I. Csiszár, J. Körner, L. Lovász, K. Marton and G. Simonyi, Entropy splitting for antiblocking corners and perfect graphs, Combinatorica 10 (1990) 27–40.
N. Dunkin, S.M. Allen, D.H. Smith and S. Hurley, Frequency assignment problems: benchmarks and lower bounds, Technical report UG-M-98–1, University of Glamorgan, 1998.
P. Erdős, Some remarks on chromatic graphs, Colloq. Math. 16 (1967) 253–256.
T. Erlebach, K. Jansen and E. Seidel, Polynomial-time approximation schemes for geometric graphs, Proceedings of the Twelth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2001), 2001, pp. 671–679.
A. Gamst, Homogeneous distribution of frequencies in a regular hexagonal cell system, IEEE Transactions on Vehicular Technology VT-31 (1982) 132–144.
A. Gamst, Some lower bounds for a class of frequency assignment problems, IEEE Transactions on Vehicular Technology VT-35 (1986) 8–14.
M.R. Garey and D.S. Johnson, Computers and Intractability, Freeman, 1979.
J.F. Georges and D.W. Mauro, Generalized vertex labelings with a condition at distance two, Congressus Numerantium 109 (1995) 141–159.
J.F. Georges, D.W. Mauro and M.I. Stein, Labelling products of complete graphs with a condition at distance two, SIAM J. Discrete Mathematics 14 (2001) 28–35.
S. Gerke, Channel assignment problems, DPhil thesis, University of Oxford, 2000.
S. Gerke, Colouring weighted bipartite graphs with a co-site constraint, Discrete Mathematics 224 (2000) 125–138.
S. Gerke and C. McDiarmid, Graph imperfection, J. Comb. Th. B, to appear.
S. Gerke and C. McDiarmid, Graph imperfection II, J. Comb. Th. B, to appear.
S. Gerke and C. McDiarmid, Channel assignment with large demands, manuscript, 2000.
S. Gerke and C. MeDiarmid, On the k-imperfection ratio of graphs, manuscript, 2001.
R.A.H. Gower and R.A. Leese, The sensitivity of channel assignment to constraint specification, in Proceedings of EMC97 Symposium, Zurich, 131–136, 1997.
A. Gräf, M. Stumpf and G. Weissenfels, On coloring unit disk graphs, Algorithmica 20 (1998) 277–293.
J.R. Griggs and D. Der-Fen Liu, The channel assignment problem for mutually adjacent sites, J. Combinatorial Theory A 68 (1994) 169–183.
J.R. Griggs and R.K. Yeh, Labelling graphs with a condition at distance two, SIAM J. Discrete Math. 5 (1995) 586–595.
M. Grötschel, L. Lovasz and A. Schrijver, Geometric Algorithms and Combinatorial Optimization, Springer-Verlag, 1988.
A. Gyárfás, Problems from the world surrounding perfect graphs, Zastosowania Matematyki Applicationes Mathematicae XIX (1987) 413–441.
W.K. Hale, Frequency assignment, Proceedings of the IEEE 68 (1980) 1497–1514.
F. Harary and M. Plantholt, Graphs whose radio coloring number equals the number of nodes, in Graph Colouring and Applications, P. Hansen and O. Marcotte editors, CRM Proceedings and Lecture Notes 23, American Mathematical Society, 1999.
F. Havet, Channel assignment and multicolouring of the induced subgraphs of the triangular lattice, Discrete Mathematics 233 (2001) 219–231.
F. Havet and J. Zerovnik, Finding a five bicolouring of a triangle-free subgraph of the triangular lattice. Discrete Mathematics, to appear.
J. van den Heuvel and S. McGuinness, Colouring the square of a planar graph, manuscript.
J. van den Heuvel, R.A. Leese and M.A. Shepherd, Graph labelling and radio channel assignment, J. Graph Theory 29 (1998) 263–283.
H.B. Hunt, M.V. Marathe, V. Radhakrishnan, S.S. Ravi, D.J. Rosenkrantz and R.E. Stearns, NC-approximation schemes for NP- and PSPACE-hard problems for geometric graphs, J. Algorithms 26 (1998) 238–274.
S. Hurley and R. Leese (editors), Models and Methods for Radio Channel Assignment, Oxford University Press, to appear.
S. Hurley, D.H. Smith and S.U. Thiel, FASoft: a system for discrete channel frequency assignment, Radio Science 32 1921–1939, 1997.
S. Janson, T. Luczak and A. Rucinski, Random Graphs, Wiley, 2000.
J. Janssen and K. Kilakos, Polyhedral analysis of channel assignment problems (part 1: tours), Research Report CDAM-96–17, LSE, August 1996.
T.R. Jensen and B. Toft, Graph Colouring Problems, Wiley, 1995.
S. Jordan and E.J. Schwabe, Worst-case performance of cellular channel assignment policies, ACM Journal of Wireless Networks 2 (1996) 265–275.
R.A. Leese, A unified approach to the assignment of radio channels on a regular hexagonal grid, IEEE Trans. Vehicular Technology 46 (1997) 968–980.
L. Lovász, Normal hypergraphs and the perfect graph conjecture, Discrete Mathematics 2 (1972) 253–267.
V.H. MacDonald, The cellular concept, The Bell System Technical Journal 58 (1979) 15–41.
E. Malesihska, S. Piskorz and G. Weißenfels, On the chromatic number of disk graphs, Networks 32 (1998) 13–22.
M.V. Marathe, H. Breu, H.B. Hunt, S.S Ravi and D.J. Rosencrantz, Simple heuristics for unit disk graphs, Networks 25 (1995) 59–68.
M.V. Marathe, V. Radhakrishnan, H.B. Hunt and S.S Ravi, Hierarchically specified unit disk graphs, Theoret Comput. Sci 174 (1997) 23–65.
T. Matsui, Approximation algorithms for maximum independent set problems and fractional coloring problems on unit disk graphs, Discrete and Computational Geometry (Tokyo, 1998) 194–200, Lecture Notes in Computer Science 1763, Springer, Berlin, 2000.
C. McDiarmid, On the chromatic number of random graphs, Random Structures and Algorithms 1 No.4 (1990) 435–442.
C. McDiarmid, A doubly cyclic channel assignment problem, Discrete Applied Mathematics 80 (1997) 263–268.
C. McDiarmid, Counting and constraint matrices, manuscript, 1998.
C. McDiarmid, Frequency-distance constraints with large distances, Discrete Applied Mathematics 223 (2000) 227–251.
C. McDiarmid, Channel assignment and graph imperfection, to appear as a chapter in Perfect Graphs, J. Ramirez and B. Reed, editors, Wiley, 2001.
C. McDiarmid, Random channel assignment in the plane, manuscript 2001.
C. McDiarmid, On the span of a random channel assignment problem, manuscript 2001.
C. McDiarmid, On the span in channel assignment problems: bounds, computing and counting, manuscript, 2001.
C. McDiarmid and B.A. Reed, Colouring proximity graphs in the plane, Discrete Mathematics 199 (1999) 123–137.
C. McDiarmid and B.A. Reed, Channel assignment and weighted colouring, Networks 36 (2000) 114–117.
C. McDiarmid and B.A. Reed, Channel assignment and bounded tree-width graphs, manuscript, 2000.
R.A. Murphy, P.M. Pardalos and M.G.C Resende, Frequency assignment problems, chapter in Handbook of Combinatorial Optimization Vol 4 (1999) (D.-Z. Dhu and P.M. Pardalos, editors).
L. Narayanan and S. Shende, Static frequency assignment in cellular networks, Proc. 4th Colloquium on Structural Information and Communications Complexity, July 1997.
J. Pach and P.K. Agarwal, Combinatorial Geometry, Wiley, 1995.
A. Paster, MSc thesis, University of Oxford, 2000.
M.D. Penrose, Random Geometric Graphs, forthcoming book.
V. Raghavan and J. Spinrad, Robust algorithms for restricted domains, Proceedings of the Twelth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2001), 2001.
A. Raychaudhuri, Intersection assignments, T-coloring, and powers of graphs, PhD thesis, Department of Mathematics, Rutgers University, NJ, 1985.
A. Raychaudhuri, Further results on T-coloring and frequency assignment problems, SIAM J. Discrete Mathematics 7 (1994) 605–613.
F.S. Roberts, T-colourings of graphs: recent results and open problems, Discrete Math. 93 (1991) 229–245.
C.A. Rogers, Packing and Covering, Cambridge University Press, 1964.
D. Sakai, Labelling chordal graphs: distance two condition, SIAM J. Discrete Math. 7 (1994) 133–140.
E.R. Scheinerman and D.H. Ullman, Fractional Graph Theory, Wiley, 1997.
N. Schnabel, S. Ubéda and J. Zerovnik, A note on upper bounds for the span of the frequency planning in celular networks, manuscript, 1999.
R. Schneider, Convex bodies: The Brunn — Minkowski Theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, 1993.
M. Shepherd, Radio Channel Assignment, DPhil thesis, University of Oxford, 1999.
G. Simonyi, Imperfection ratio and graph entropy, manuscript, 1999.
D.H. Smith and S. Hurley, Bounds for the frequency assignment problem, Discrete Mathematics 167/168 (1997) 571–582.
D.H. Smith, S.M. Allen and S. Hurley, Lower bounds for channel assignment, in S. Hurley and R. Leese (editors), Models and Methods for Radio Channel Assignment, Oxford University Press, to appear.
C.-E. Sundberg, Alternative cell configurations for digital mobile radio systems, The Bell System Technical Journal 62 (1983) 2037–2065.
B.A. Tesman, Applications of forbidden distance graphs to T-colorings, Congressus Numerantium 74 (1990) 15–24.
W.J. Watkins, S. Hurley and D.H. Smith, Evaluation of models for area coverage, Report to UK Radiocommunications Agency, Department of Computer Science, Cardiff University, December 1998.
W. Webb, The Complete Wireless Communications Professional, Artech House Publishers, 1999.
D.J.A. Welsh and G. Whittle, Arrangements, channel assignment, and associated polynomials, Adv. in Appl. Math 23 (1999) 375–406.
D.B. West, Introduction to Graph Theory, Prentice Hall, 1996.
R.M. Whitaker, S. Hurley and D.H. Smith, Frequency assignment heuristics for area coverage problems, Report to UK Radiocommunications Agency, Department of Computer Science, Cardiff University, September 2000.
X. Zhu, Circular chromatic number: a survey, Discrete Math. 229 (2001) 371–410.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag New York, Inc.
About this chapter
Cite this chapter
McDiarmid, C. (2003). Discrete Mathematics and Radio Channel Assignment. In: Reed, B.A., Sales, C.L. (eds) Recent Advances in Algorithms and Combinatorics. CMS Books in Mathematics / Ouvrages de mathématiques de la SMC. Springer, New York, NY. https://doi.org/10.1007/0-387-22444-0_2
Download citation
DOI: https://doi.org/10.1007/0-387-22444-0_2
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4684-9268-2
Online ISBN: 978-0-387-22444-2
eBook Packages: Springer Book Archive