Publication: Coloring fuzzy graphs
Loading...
Full text at PDC
Publication Date
2005-06
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
Pergamon Elsevier Science
Citations
Exportar
Abstract
Given a graph G = (V, E), a coloring function C assigns an integer value C(i) to each node i epsilon V in such a way that the extremes of any edge {i,j} epsilon E cannot share the same color, i.e., C(i) epsilon C(j). Two different approaches to the graph coloring problem of a fuzzy graph 6 = ( V, (E) over tilde) are introduced in this paper. The classical concept of the (crisp) chromatic number of a graph is generalized for these approaches. The first approach is based on the successive coloring functions C-x of the crisp graphs G(x) = (T E.), the alpha-cuts of (G) over tilde; the traffic lights problem is analyzed following this approach. The second approach is based on an extension of the concept of coloring function by means of a distance defined between colors; a timetabling problem is analyzed within this approach. An exact algorithm for obtaining the chromatic number associated with the second approach is proposed, and some computational results on randomly generated fuzzy graphs are reported.
Description
UCM subjects
Unesco subjects
Keywords
Citation
Garey MR, Johnson DS. Computers and intractability. A guide to the theory of NP—completeness. New York: W.H. Freeman and Company; 1979.
Liu CL. Introduction to combinatorial mathematics. New York: Mc-Graw Hill; 1968.
Eilon S, Christo4des N. The loading problem. Management Science 1971;17(5):259–68.
Christofides N. Graph theory. An algorithmic approach. London: Academic Press; 1975.
Roberts FS. On the mobile radio frequency assignment problem and the traffic light phasing problem. Annals of the New York Academy of Sciences 1979; 319:466-83.
Pardalos PM, Mavridou T, Xue J. The graph coloring problem: a bibliographic survey. In: Du DZ, Pardalos PM, editors. Handbook of combinatorial optimization, vol. 2. Boston: Kluwer Academic Publishers; 1998. p. 331–95.
Chaitin GJ, Auslander MA, Chandra AK, Cocke J, Hopkins ME, Markstein PW. Register allocation via coloring. Computer Languages 1981;6(1):47–57.
Yáñez J, Ramírez J. The robust coloring problem. European Journal of Operational Research 2003;148(3):546– 58.
Zadeh LA. Similarity relations and fuzzy ordering. Information Sciences 1971;3(2):177–200.
Zadeh LA. The concept of a linguistic variable and its application to approximate reasoning-I. Information Sciences 1975;8(3):199–249; Zadeh LA. The concept of a linguistic variable and its application to approximate reasoning—I. Information Sciences 1975;8(4):301-57 Zadeh LA. The concept of a linguistic variable and its application to approximate reasoning—I. Information Sciences 1975;9(1):43–80.
Cutello V, Montero J, Yáñez J. Structure functions with fuzzy states. Fuzzy Sets and Systems 1996;83(2):189–2002.
Kaufmann A. Introduction Xa la théorie des sous-ensembles Nous, 1o Eléments théoriques de base. Paris: Masson et Cie; 1976.
Rosenfeld A. Fuzzy graphs. In: Zadeh LA, Fu KS, Shimura M, editors. Fuzzy sets and their applications to cognitive and decision processes. New York: Academic Press; 1975.p.77-95.
Kóczy LT. Fuzzy graphs in the evaluation and optimization of networks. Fuzzy Sets and Systems 1992;46(3):307–19.
Ramírez J. Extensiones del problema de coloración de grafos. PhD thesis. Universidad Complutense de Madrid, Madrid; 2001.
Delgado M, Verdegay JL, Vila MA. On valuation and optimization problems in fuzzy graphs: a general approach and some particular cases. ORSA Journal on Computing 1990;2 (1) 74-83.
Golumbic MC. Algorithmic graph theory and perfect graphs. New York: Academic Press; 1980.
Opsut RJ, Roberts FS. I-colorings, I-phasings, and I-intersection assignments for graphs, and their applications.Networks 1983;13:327-45.
Stoffers KE. Scheduling of tra6c lights. A new approach. Transportation Research 1968;2:199–234.
Kerre EE. Basic principles of fuzzy set theory for the representation and manipulation of imprecision anduncertainty. In: Kerre EE, editor. Introduction to the basic principles of fuzzy set theory and some of its applications, 2nd revised edition. Gent: Communication and Cognition; 1993. p. 57–70.
Korman SM. The graph colouring problem. In: Christofides N, Mingozzi A, Toth P, Sandi C, editors. Combinatorial optimization. New York:Wiley. 1979. p.211-35.
Sager TJ, Lin SJ. A pruning procedure for exact graph coloring. ORSA Journal on Computing 1991;3(3):226–30.
Mehrotra A, Trick MA. A column generation approach for graph coloring. INFORMS Journal on Computing 1996;8(4):344–54.
Sewell EC. An improved algorithm for exact graph coloring. In: Johnson DS, Trick MA, editors. DIMACS series in discrete mathematics and theoretical computer science, vol.
Providence, RI: American Mathematical Society; 1996. p. 359–73. [25] Herrmann F, Hertz A. Finding the chromatic number by means of critical graphs. Journal of Experimental Algorithmics 2002; 7(Article 10):1–9.
Eppstein D. Small maximal independent sets and faster exact graph coloring. Journal of Graph Algorithms and Applications 2003;7(2):131–40.
Brélaz D. New methods to color the vertices of a graph. Communications of the ACM 1979;22(4):251–6.
Avanthay C, Hertz A, Zu3erey N. A variable neighborhood search for graph coloring. European Journal of Operational Research 2003;151(2):379–88.
di Blas A, Jagota A, Hughey R. A range-compaction heuristic for graph coloring. Journal of Heuristics 2003;9(6):489–506.
Bullnheimer B. An examination scheduling model to maximize student’s study time. In: Burke E, Ross P, editors.Practic and theory of automated timetabling II. Berlin: Springer; 1998.p. 78-91.
Christo4des N. An algorithm for the chromatic number of a graph. The Computer Journal 1971;14(1):38–9.
Brown JR. Chromatic scheduling and the chromatic number problem. Management Science 1972;19(4):456–63.
PeemYoller J. A correction to Br)elaz’s modi4cation of Brown’s coloring algorithm. Communications of the ACM 1983; 26 (8): 595-7.
Kubale M, Jackowski B. A generalized implicit enumeration algorithm for graph coloring. Communications of the ACM 1985(4):412-8.