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Est-il vrai qu’il existe une constante c telle que, pour tout graphe planaire 2 connexe G,

Dans le document Colorations de graphes et applications (Page 196-200)

Largeur arborescente linéaire des graphes planaires extérieurs

Problème 13. Est-il vrai qu’il existe une constante c telle que, pour tout graphe planaire 2 connexe G,

1 2pw(G

)− c ≤ pw(G) ≤ 2pw(G) + c?

Si la réponse est affirmative, le facteur multiplicatif deux serait optimal, d’après le théo- rème 70.

Bibliographie

AH77. K. Appel and W. Haken. The solution of the four-color-map problem. Sci. Amer., 237(4) :108–121, 152, 1977.

AH89. K. Appel and W. Haken. Every planar map is four colorable, volume 98 of Contem-

porary Mathematics. American Mathematical Society, Providence, RI, 1989. With

the collaboration of J. Koch.

AHL02. N. Alon, S. Hoory, and N. Linial. The Moore bound for irregular graphs. Graphs

Combin., 18(1) :53–57, 2002.

Alo00. N. Alon. Degrees and choice numbers. Random Structures Algorithms, 16(4) :364– 368, 2000.

Arc86. D. Archdeacon. Coverings of graphs by cycles. In Proceedings of the seventeenth

Southeastern international conference on combinatorics, graph theory, and com- puting, volume 53, pages 7–14, 1986.

Arc95. D. Archdeacon. Generalizations of tait coloring of cubic graphs, Problems

in Topological Graph Theory. http://www.emba.uvm.edu/~archdeac/

problems/petecol.htm, 1995.

AT92. N. Alon and M. Tarsi. Colorings and orientations of graphs. Combinatorica, 12(2) :125–134, 1992.

AZ98. N. Alon and A. Zaks.T -choosability in graphs. Discrete Appl. Math., 82(1-3) :1–

13, 1998.

BF02. H. L. Bodlaender and F. V. Fomin. Approximation of pathwidth of outerplanar graphs. J. Algorithms, 43(2) :190–200, 2002.

BFFS02. L. Barrière, P. Flocchini, P. Fraigniaud, and N. Santoro. Capture of an intruder by mobile agents. In 14th ACM Symp. on Parallel Algorithms and Architectures (SPAA

’02), Winnipeg, August 10-13, 2002.

BK96. H. L. Bodlaender and T. Kloks. Efficient and constructive algorithms for the path- width and treewidth of graphs. J. Algorithms, 21(2) :358–402, 1996.

BK98. H. Breu and D. G. Kirkpatrick. Unit disk graph recognition is NP-hard. Com-

put. Geom., 9(1-2) :3–24, 1998.

BMT03. V. Bouchitté, F. Mazoit, and I. Todinca. Chordal embeddings of planar graphs.

Discrete Math., 273(1-3) :85–102, 2003. EuroComb’01 (Barcelona).

Bod96. H. L. Bodlaender. A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput., 25(6) :1305–1317, 1996.

Bod98. H. L. Bodlaender. A partialk-arboretum of graphs with bounded treewidth. Theor. Comput. Sci., 209(1-2) :1–45, 1998.

Bor77. O. V. Borodin. Criterion of chromaticity of a degree prescription (in Russian). In

Abstracts of IV All-Union Conf. on Theoretical Cybernetics (Novosibirsk), pages

127–128, 1977.

Bor84. O. V. Borodin. Solution of the Ringel problem on vertex-face coloring of planar graphs and coloring of 1-planar graphs. Metody Diskret. Analiz., 41 :12–26, 108,

1984.

Bor92. O. V. Borodin. Cyclic coloring of plane graphs. Discrete Math., 100(1-3) :281–289, 1992. Special volume to mark the centennial of Julius Petersen’s “Die Theorie der regulären Graphs”, Part I.

Bor95. O. V. Borodin. A new proof of the6 color theorem. J. Graph Theory, 19(4) :507–

521, 1995.

Bre67. R. L. Breisch. An intuitive approach to speleotopology. Southwestern Cavers, 6 :72–78, 1967.

Bro41. R. L. Brooks. On colouring the nodes of a network. Proc. Cambridge Philos. Soc., 37 :194–197, 1941.

BS91. D. Bienstock and P. Seymour. Monotonicity in graph searching. J. Algorithms, 12(2) :239–245, 1991.

BSZ99. O. V. Borodin, D. P. Sanders, and Y. Zhao. On cyclic colorings and their generali- zations. Discrete Math., 203(1-3) :23–40, 1999.

CCJ90. B. N. Clark, C. J. Colbourn, and D. S. Johnson. Unit disk graphs. Discrete Math., 86(1-3) :165–177, 1990.

CCW86. L. J. Cowen, R. H. Cowen, and D. R. Woodall. Defective colorings of graphs in sur- faces : partitions into subgraphs of bounded valency. J. Graph Theory, 10(2) :187– 195, 1986.

CGJ97. L. Cowen, W. Goddard, and C. E. Jesurum. Defective coloring revisited. J. Graph

Theory, 24(3) :205–219, 1997.

CHS06a. R. Corrêa, F. Havet, and J.-S. Sereni. Around a brooks’ like theorem for improper colouring. In preparation, 2006.

CHS06b. D. Coudert, F. Huc, and J.-S. Sereni. Pathwidth of outerplanar graphs. Journal of

Graph Theory, 2006. To appear.

CHS06c. D. Coudert, F. Huc, and J.S. Sereni. Pathwidth of outerplanar graphs. Research report, INRIA Research Report 5804 and I3S Research Report I3S/RR-2006-02- FR, January 2006.

CPPS05. D. Coudert, S. Pérennes, Q.-C. Pham, and J.-S. Sereni. Rerouting requests inWDM networks. In Septièmes Rencontres Francophones sur les Aspects Algorithmiques

des Télécommunications (AlgoTel’05), pages 17–20, Presqu’île de Giens, May

2005.

CS06. D. Coudert and J.-S. Sereni. Graphs with small process number. In preparation, 2006.

Die05. R. Diestel. Graph theory, volume 173 of Graduate Texts in Mathematics. Springer- Verlag, Berlin, third edition, 2005.

Bibliographie 185

DJL93. A. Dessmark, K. Jansen, and A. Lingas. The maximum k-dependent and f -

dependent set problem. In Algorithms and computation (Hong Kong, 1993), vo- lume 762 of Lecture Notes in Comput. Sci., pages 88–97. Springer, 1993.

DKL87. N. Deo, S. Krishnamoorthy, and M. A. Langston. Exact and approximate solutions for the gate matrix layout problem. IEEE Transactions on Computer-Aided Design, 6 :79–84, 1987.

DMGG96. P. Danziger, E. Mendelsohn, M. J. Grannell, and T. S. Griggs. Five-line configura- tions in Steiner triple systems. Utilitas Math., 49 :153–159, 1996.

DPS02. J. Díaz, J. Petit, and M. Serna. A survey of graph layout problems. ACM Computing

Surveys, 34(3) :313–356, 2002.

DŠT05. Z. Dvoˇrák, R. Škrekovski, and M. Tancer. List-colouring squares of sparse subcubic graphs. Technical Report IMFM-(2005)-PS-985, University of Ljubljana, Slovenia, 2005.

Edm65. J. Edmonds. Minimum partition of a matroid into independent subsets. J. Res. Nat.

Bur. Standards Sect. B, 69B :67–72, 1965.

EH99. N. Eaton and T. Hull. Defective list colorings of planar graphs. Bull. Inst. Com-

bin. Appl., 25 :79–87, 1999.

ERT80. P. Erd˝os, A. L. Rubin, and H. Taylor. Choosability in graphs. In Proceedings of the

West Coast Conference on Combinatorics, Graph Theory and Computing (Hum- boldt State Univ., Arcata, Calif., 1979), Congress. Numer., XXVI, pages 125–157,

Winnipeg, Man., 1980. Utilitas Math.

EST94. J. A. Ellis, I. H. Sudborough, and J. S. Turner. The vertex separation and search number of a graph. Inform. and Comput., 113(1) :50–79, 1994.

EWHK98. T. Emden-Weinert, S. Hougardy, and B. Kreuter. Uniquely colourable graphs and the hardness of colouring graphs of large girth. Combin. Probab. Comput., 7(4) :375–386, 1998.

FKK96. U. Fößmeier, G. Kant, and M. Kaufmann. 2-visibility drawings of plane graphs. In Proceedings of Graph Drawing ’96, volume 1190 of Lecture Notes in Comput.

Sci., pages 155–168. Springer, Berlin, 1996.

Fom03. F. V. Fomin. Pathwidth of planar and line graphs. Graphs and Combinatorics, 19(1) :91–99, 2003.

FR94. G. Fan and A. Raspaud. Fulkerson’s conjecture and circuit covers. J. Combin.

Theory Ser. B, 61(1) :133–138, 1994.

FT06. F. Fomin and D. M. Thilikos. On self duality of pathwidth in polyhedral graph embeddings. Report in Informatics 316, University of Bergen, March 2006. Ful71. D. R. Fulkerson. Blocking and anti-blocking pairs of polyhedra. Math. Program-

ming, 1 :168–194, 1971.

Gam86. A. Gamst. Some lower bounds for a class of frequency assignment problems. IEEE

Transactions on Vehicular Technology, 35 :8–14, 1986.

GJ79. M. R. Garey and D. S. Johnson. Computers and intractability. W. H. Freeman and Co., San Francisco, Calif., 1979. A guide to the theory of NP-completeness, A Series of Books in the Mathematical Sciences.

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