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Towards a Theory of Geometric Graphs
Edited by: János Pach, City College, City University of New York, NY, and Hungarian Academy of Sciences, Budapest, Hungary
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Contemporary Mathematics
2004; 283 pp; softcover
Volume: 342
ISBN-10: 0-8218-3484-3
ISBN-13: 978-0-8218-3484-8
List Price: US$87 Member Price: US$69.60
Order Code: CONM/342

The early development of graph theory was heavily motivated and influenced by topological and geometric themes, such as the Königsberg Bridge Problem, Euler's Polyhedral Formula, or Kuratowski's characterization of planar graphs. In 1936, when Dénes König published his classical Theory of Finite and Infinite Graphs, the first book ever written on the subject, he stressed this connection by adding the subtitle Combinatorial Topology of Systems of Segments. He wanted to emphasize that the subject of his investigations was very concrete: planar figures consisting of points connected by straight-line segments. However, in the second half of the twentieth century, graph theoretical research took an interesting turn. In the most popular and most rapidly growing areas (the theory of random graphs, Ramsey theory, extremal graph theory, algebraic graph theory, etc.), graphs were considered as abstract binary relations rather than geometric objects. Many of the powerful techniques developed in these fields have been successfully applied in other areas of mathematics. However, the same methods were often incapable of providing satisfactory answers to questions arising in geometric applications.

In the spirit of König, geometric graph theory focuses on combinatorial and geometric properties of graphs drawn in the plane by straight-line edges (or more generally, by edges represented by simple Jordan arcs). It is an emerging discipline that abounds in open problems, but it has already yielded some striking results which have proved instrumental in the solution of several basic problems in combinatorial and computational geometry. The present volume is a careful selection of 25 invited and thoroughly refereed papers, reporting about important recent discoveries on the way Towards a Theory of Geometric Graphs.

Graduate students and research mathematicians interested in discrete mathematics.

• H. Alt, C. Knauer, G. Rote, and S. Whitesides -- On the complexity of the linkage reconfiguration problem
• G. Arutyunyants and A. Iosevich -- Falconer conjecture, spherical averages and discrete analogs
• P. Brass -- Turán-type extremal problems for convex geometric hypergraphs
• G. Cairns, M. McIntyre, and Y. Nikolayevsky -- The thrackle conjecture for $$K_5$$ and $$K_{3,3}$$
• V. Dujmović and D. R. Wood -- Three-dimensional grid drawings with sub-quadratic volume
• A. Dumitrescu and R. Radoičić -- On a coloring problem for the integer grid
• D. Eppstein -- Separating thickness from geometric thickness
• R. E. Jamison -- Direction trees in centered polygons
• A. Kaneko, M. Kano, and K. Suzuki -- Path coverings of two sets of points in the plane
• G. O. H. Katona, R. Mayer, and W. A. Woyczynski -- Length of sums in a Minkowski space
• N. H. Katz and G. Tardos -- A new entropy inequality for the Erdős distance problem
• A. Kostochka -- Coloring intersection graphs of geometric figures with a given clique number
• L. Lovász, K. Vesztergombi, U. Wagner, and E. Welzl -- Convex quadrilaterals and $$k$$-sets
• H. Maehara -- Distance graphs and rigidity
• J. Nešetřil, J. Solymosi, and P. Valtr -- A Ramsey property of planar graphs
• J. Pach, R. Radoičić, and G. Tóth -- A generalization of quasi-planarity
• J. Pach and M. Sharir -- Geometric incidences
• M. A. Perles and R. Pinchasi -- Large sets must have either a $$k$$-edge or a $$(k+2)$$-edge
• R. Pinchasi and R. Radoičić -- Topological graphs with no self-intersecting cycle of length 4
• I. Z. Ruzsa -- A problem on restricted sumsets
• F. Shahrokhi, O. Sýkora, L. A. Székely, and I. Vrťo -- The gap between crossing numbers and convex crossing numbers
• J. Solymosi and V. Vu -- Distinct distances in high dimensional homogeneous sets
• J. Spencer -- The biplanar crossing number of the random graph
• K. J. Swanepoel and P. Valtr -- The unit distance problem on spheres
• L. Székely -- Short proof for a theorem of Pach, Spencer, and Tóth