By F. P. Ramsey (auth.), Ira Gessel, Gian-Carlo Rota (eds.)

This quantity surveys the advance of combinatorics because 1930 by means of offering in chronological order the elemental result of the topic proved in over 5 a long time of unique papers by:.-T. van Aardenne-Ehrenfest.-R.L. Brooks.-N.G. de Bruijn.-G.F. Clements.-H.H. Crapo.-R.P. Dilworth.-J. Edmonds.-P.Erdös.-L.R. Ford, Jr.-D.R. Fulkerson.-D. Gale.-L. Geissinger.-I.J. Good.-R.L. Graham.-A.W. Hales.-P. Hall.-P.R. Halmos.-R.I. Jewett.-I. Kaplansky.-P.W. Kasteleyn.-G. Katona.-D.J. Kleitman.-K. Leeb.-B. Lindström.-L. Lovász.-D. Lubell.-C. St. J.A. Nash-Williams.-G. Pólya.-F.P. Ramsey.-G.C. Rota.-B.L. Rothschild.-H.J. Ryser.-C. Schensted.-M.P. Schützenberger.-R.P. Stanley.-G. Szekeres.-W.T. Tutte.-H.E. Vaughan.-H. Whitney.

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**Example text**

A planar graph may be mapped on a plane so that any desired regioll is the outside region. We map the graph on a sphere, and rotate it so that the north pole lie! inside the given region. By stereographic projection, the graph is mappec onto the plane so that the given region is the outside region. We return now to the work in hand. 41 356 HASSLER WHITNEY THEOREM [April 27. If the components of a graph G are planar, G is planar. Suppose the graphs Gi and G2 are planar, and G' is formed by letting the vertices ai and a2 of Gi and G2 coalesce.

Map Gi on a sphere, and map G2 on a plane so that one of the regions adjacent to the vertex a2 is the outside region. Shrink the portion of the plane containing G2 so it will fit into one of the regions of Gi adjacent to ai. * The theorem follows as a repeated application of this process. THEOREM 28. Let G and G' be dual graphs, and let a(ab), a'(a'b') be two corresponding arcs. Form Gdrom G by dropping out the arc a(ab), and form G{ from G' by dropping out the arc a'(a'b'), and letting the vertices a' and b' coalesce if they are not already the same vertex.

Xl' x~, ••• , Xn). But this is a formula of the type previously dealt with, except for the variable propositions p, q, ... , which are easily eliminated by considering the different cases of their truth and falsity, the formula being consistent jf it is cOl1sist-ent in one snch casco Reprinted from Proc. London Math. Soc. 30 (1930),264-286 24 NON-SEPARABLE AND PLANAR GRAPHS* BY HASSLER WHITNEY Introduction. In this paper the structure of graphs is studied by purely combinatorial methods. The concepts of rank and nullity are fundamental.