Distinct distances in graph drawings

Electronic Journal of Combinatorics

Volumn 15, Issue 1, 2008, Pages

  Carmi, Paz ^a   Dujmović, Vida ^a   Morin, Pat ^a   Wood, David R ^b  

^a CARLETON UNIVERSITY   (Canada)

^b UNIVERSITY OF MELBOURNE   (Australia)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 50249127773     PISSN: None     EISSN: 10778926     Source Type: Journal    
DOI: 10.37236/831     Document Type: Article

References (53)

1. E. ALTMAN. On a problem of P. Erdocombining double acute accents. Amer. Math. Monthly, 70:148-157, 1963.
2. E. ALTMAN. Some theorems on convex polygons. Canad. Math. Bull., 15:329-340, 1972.
3. DAN ARCHDEACON. Planar graphs with integer length straight edges, 1995. http://www.emba.uvm.edu/~archdeac/problems/integer.htm.
4. JÁNOS BARÁT, JIŘŔ MATOUŠEK, AND DAVID R. WOOD. Bounded-degree graphs have arbitrarily large geometric thickness. Electron. J. Combin., 13(1):R3, 2006.
5. MARIA BELK. Realizability of graphs in three dimensions. Discrete Comput. Geom., 37:139-162, 2007.
6. MARIA BELK AND ROBERT CONNELLY. Realizability of graphs. Discrete Comput. Geom., 37:125-137, 2007.
7. EDWARD A. BENDER AND E. RODNEY CANFIELD. The asymptotic number of labeled graphs with given degree sequences. J. Combin. Theory Ser. A, 24:296-307, 1978.
8. HANS L. BODLAENDER. A partial k-arboretum of graphs with bounded treewidth. Theoret. Comput. Sci., 209(1-2):1-45, 1998.
9. JULIA BÖTTCHER, KLAAS P. PRUESSMANN, ANUSCH TARAZ, AND ANDREAS WÜRFL. Bandwidth, treewidth, separators, expansion, and universality. In Proc. Topological and Geometric Graph Theory (TGGT '08). 2008.
10. FRED BUCKLEY AND FRANK HARARY. On the Euclidean dimension of a wheel. Graphs Combin., 4(1):23-30, 1988.
11. WAI HONG CHAN, PETER C. B. LAM, AND WAI CHEE SHIU. Cyclic bandwidth with an edge added. Discrete Appl. Math., 156(1):131-137, 2008.
12. GUOLI DING AND BOGDAN OPOROWSKI. Some results on tree decomposition of graphs. J. Graph Theory, 20(4):481-499, 1995.
13. VIDA DUJMOVIĆ, DAVID EPPSTEIN, MATTHEW SUDERMAN, AND DAVID R. WOOD. Drawings of planar graphs with few slopes and segments. Comput. Geom., 38:194-212, 2007.
14. VIDA DUJMOVIĆ, MATTHEW SUDERMAN, AND DAVID R. WOOD. Graph drawings with few slopes. Comput. Geom., 38:181-193, 2007.
15. PAUL ERDOCOMBINING DOUBLE ACUTE ACCENTS. On sets of distances of n points. Amer. Math. Monthly, 53:248-250, 1946.
16. PAUL ERDOCOMBINING DOUBLE ACUTE ACCENTS, FRANK HARARY, AND WILLIAM T. TUTTE. On the dimension of a graph. Mathematika, 12:118-122, 1965.
17. PETER FRANKL AND RICHARD M. WILSON. Intersection theorems with geometric consequences. Combinatorica, 1(4):357-368, 1981.
18. JIM GEELEN, ANJIE GUO, AND DAVID MCKINNON. Straight line embeddings of cubic planar graphs with integer edge lengths. J. Graph Theory, 58(3):270-274, 2008.
19. HUGO HADWIGER AND HANS DEBRUNNER. Combinatorial geometry in the plane. Holt, New York, 1964.
20. FRANK HARARY, PAUL C. KAINEN, AND ADRIAN RISKIN. Every graph of cyclic bandwidth 3 is toroidal. Bull. Inst. Combin. Appl., 27:81-84, 1999.
21. HEIKO HARBORTH AND ARNFRIED KEMNITZ. Integral representations of graphs. In Contemporary methods in graph theory, pp. 359-367. Bibliographisches Inst., Mannheim, 1990.
22. HEIKO HARBORTH, ARNFRIED KEMNITZ, MEINHARD MÖLLER, AND ANDREAS SÜSSENBACH. Ganzzahlige planare Darstellungen der platonischen Körper. Elem. Math., 42(5):118-122, 1987.
23. ROBERT HOCHBERG AND PAUL O'DONNELL. Some 4-chromatic unit-distance graphs without small cycles. Geombinatorics, 5(4):137-141, 1996.
24. BORIS HORVAT AND TOMAŽ PISANSKI. Products of unit distance graphs. In 6th Slovenian International Conference on Graph Theory. Bled, Slovenia, 2007.
25. NETS HAWK KATZ AND GÁBOR TARDOS. A new entropy inequality for the Erdocombining double acute accents distance problem. In Towards a theory of geometric graphs, vol. 342 of Contemp. Math., pp. 119-126. Amer. Math. Soc., 2004.
26. ARNFRIED KEMNITZ AND HEIKO HARBORTH. Plane integral drawings of planar graphs. Discrete Math., 236(1-3):191-195, 2001.
27. BALÁZS KESZEGH, DÖMÖTÖR PÁLVÖLGYI, JÁNOS PACH, AND GÉZA TÓTH. Drawing cubic graphs with at most five slopes. Comput. Geom., 40(2):138-147, 2008.
28. PETER C. B. LAM, WAI CHEE SHIU, AND WAI HONG CHAN. Characterization of graphs with equal bandwidth and cyclic bandwidth. Discrete Math., 242(1-3):283-289, 2002.
29. DAVID G. LARMAN AND C. AMBROSE ROGERS. The realization of distances within sets in Euclidean space. Mathematika, 19:1-24, 1972.
30. YI XUN LIN. The cyclic bandwidth problem. Systems Sci. Math. Sci., 7(3):282-288, 1994.
31. HOOI-TONG LOH AND HOON HENG TEH. On the dimension of product graphs. Nanta Math., 1:68-71, 1966/1967.
32. HIROSHI MAEHARA. Note on induced subgraphs of the unit distance graph En. Discrete Comput. Geom., 4(1):15-18, 1989.
33. HIROSHI MAEHARA AND VOJTĚCH RÖDL. On the dimension to represent a graph by a unit distance graph. Graphs Combin., 6(4):365-367, 1990.
34. BRENDAN D. MCKAY. Asymptotics for symmetric 0-1 matrices with prescribed row sums. Ars Combin., 19(A):15-25, 1985.
35. PAUL O'DONNELL. Arbitrary girth, 4-chromatic unit distance graphs in the plane. I. Graph description. Geombinatorics, 9(3):145-152, 2000.
36. PAUL O'DONNELL. Arbitrary girth, 4-chromatic unit distance graphs in the plane. II. Graph embedding. Geombinatorics, 9(4):180-193, 2000.
37. JÁNOS PACH AND PANKAJ K. AGARWAL. Combinatorial geometry. John Wiley & Sons Inc., New York, 1995.
38. JÁNOS PACH AND DÖMÖTÖR PÁLVÖLGYI. Bounded-degree graphs can have arbitrarily large slope numbers. Electron. J. Combin., 13(1):N1, 2006.
39. DAN PRITIKIN. All unit-distance graphs of order 6197 are 6-colorable. J. Combin. Theory Ser. B, 73(2):159-163, 1998.
40. A. M. RAǏGORODSKIǏ. On the chromatic number of a space. Uspekhi Mat. Nauk, 55(2):147-148, 2000.
41. BRUCE A. REED. Algorithmic aspects of tree width. In BRUCE A. REED AND CLÁUDIA L. SALES, eds., Recent Advances in Algorithms and Combinatorics, pp. 85-107. Springer, 2003.
42. MICHAEL REID. On the connectivity of unit distance graphs. Graphs Combin., 12(3):295-303, 1996.
43. LAJOS RÓNYAI, LÁSZLÓ BABAI, AND MURALI K. Ganapathy. On the number of zero-patterns of a sequence of polynomials. J. Amer. Math. Soc., 14(3):717-735, 2001.
44. PETRA SCHEFFLER. Die Baumweite von Graphen als ein Maß für die Kompliziertheit algorithmischer Probleme. Ph.D. thesis, Akademie der Wissenschaften der DDR, Berlin, Germany, 1989.
45. SAHARON SHELAH AND ALEXANDER SOIFER. Axiom of choice and chromatic number of the plane. J. Combin. Theory Ser. A, 103(2):387-391, 2003.
46. JÓZSEF SOLYMOSI, GÁBOR TARDOS, AND CSABA D. TÓTH. The k most frequent distances in the plane. Discrete Comput. Geom., 28(4):639-648, 2002.
47. JÓZSEF SOLYMOSI AND CSABA D. TÓTH. Distinct distances in the plane. Discrete Comput. Geom., 25(4):629-634, 2001.
48. JOEL H. SPENCER, ENDRE SZEMERÉDI, AND WILLIAM T. TROTTER, JR. Unit distances in the Euclidean plane. In BÉLA BOLLOBAS, ed., Graph theory and combinatorics, pp. 293-303. Academic Press, London, 1984.
49. LÁSZLÓ A. SZÉKELY. Crossing numbers and hard Erdocombining double acute accents problems in discrete geometry. Combin. Probab. Comput., 6(3):353-358, 1997.
50. GÁBOR TARDOS. On distinct sums and distinct distances. Adv. Math., 180(1):275-289, 2003.
51. DAVID R. WOOD. On tree-partition-width, 2007. arXiv.org/math/0602507.
52. NICHOLAS WORMALD. Some problems in the enumeration of labelled graphs. Ph.D. thesis, University of Newcastle, United Kingdom, 1978.
53. SANMING ZHOU. Bounding the bandwidths for graphs. Theoret. Comput. Sci., 249(2):357-368, 2000.