Abstract
Consider a mining project where four substances sand, turf, coal, and earth are being extracted at five different sources. One also has four locations where the different materials will be collected, labelled S, T, C, and E respectively.
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References
V. B. Alekseev and V. S. Gonchakov, Thickness of arbitrary complete graphs, Mat. Sbornik (143) 101 (1976), 212–230.
L. W. Beineke, Complete bipartite graphs: decomposition into planar subgraphs, A Seminar in Graph Theory, Holt, Rinehart and Winston, New York, 1967, 42-53.
L. W. Beineke, The decomposition of complete graphs into planar subgraphs, Graph Theory and Theoretical Physics, Academic Press, New York, 1967, 139-154.
L. W. Beineke and F. Harary, The thickness of the complete graph, Canad. J. Math. 17 (1965), 850–859.
L. W. Beineke, F. Harary, and J. W. Moon, On the thickness of the complete bipartite graph, Proc. Camb. Phil. Soc. 60 (1964), 1–5.
J. Blazek and M. Koman, A minimal problem concerning complete plane graphs, Theory of Graphs and its Applications, Academic Press, New York, 1964, 113–177.
M. Gardner, Mathematical Games, Scientific American, February 1980.
W. Gustin, Orientable embedding of Cayley graphs, Bull. Amer. Math. Soc. 69 (1963), 272–275.
R. K. Guy, A combinatorial problem, Bull. Malayan Math. Soc. 7 (1960), 68–72.
R. K. Guy, Crossing number of graphs, Graph Theory and Applications, Springer-Verlag, New York, 1972, 111–124.
P. J. Heawood, Map Colour Theorem, Quart. J. Math. 24 (1890), 332–338.
L. Heffter, Über das Problem der Nachbargebiete, Math. Ann. 38 (1891), 477–508.
B. Jackson and G. Ringel, The splitting number of the complete bipartite graph, to appear in Archiv der Math.
B. Jackson and G. Ringel, The splitting number of the complete graph, in preparation.
B. Jackson and G. Ringel, Heawood’s empire problem on the plane, to appear in J. reine angew. Math.
D. J. Kleitman, The crossing number of K5,n, J. Combinatorial Theory 9 (1970), 315–323.
J. Vasak, The thickness of the complete graph having 6m+4 points, to appear.
K. Zarankiewicz, On a problem of P. Turán concerning graphs, Fund. Math. 41 (1954), 137–145.
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© 1984 Springer-Verlag Berlin Heidelberg
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Jackson, B., Ringel, G. (1984). Plane Constructions for Graphs, Networks, and Maps Measurements of Planarity. In: Hammer, G., Pallaschke, D. (eds) Selected Topics in Operations Research and Mathematical Economics. Lecture Notes in Economics and Mathematical Systems, vol 226. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45567-4_24
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DOI: https://doi.org/10.1007/978-3-642-45567-4_24
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