By Gian-Carlo Rota, Ira Gessel
This quantity surveys the improvement of combinatorics due to the fact that 1930 via providing in chronological order the elemental result of the topic proved in over 5 a long time of unique papers by way of: 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.- okay. Leeb.- B. Lindström.- L. Lovász.- D. Lubell.- C. St. J.A. Nash-Williams.- G. Pólya.-R. Rado.- 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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Then the following four statements are all equivalent: (1) GI , . . , Gm are the components of G. (2) No two of the graphs Gh . . , Gm have an arc in common, and there is no circuit in G containing arcs of more than one of these graphs. (3) No subset of these graphs form a circuit of graphs. (4) If R, R I , • • • , Rm are the ranks of G, GI , . . , Gm respectively, then R = RI + 0 0 0 + Rm o We note that we cannot replace the word rank by the word nullity in (4). For let G be the graph containing the arcs ex (ab) , (3(ab) , 'Y(ab).
As a special case of this theorem, we have THEOREM 26. A dual of a non-separable graph is non-separable. 9. Planar graphs. Up till now, we have been considering abstract graphs alone. However, the definition of a planar graph is topological in character. This section may be considered as an application of the theory of abstract graphs to the theory of topological graphs. Definitions. A topological graph is called planar if it can be mapped in a (1, 1) continuous manner on a sphere (or a plane).
Let H' be any sub graph of G' , and let H be the complement of the corresponding sub graph of G. Then, as G' is a dual of G, * While this definition agrees with the ordinary one for graphs lying on a plane or sphere, a graph on a surface of higher connectivity, such as the torus, has in general no dual. ) 37 352 HASSLER WHITNEY r' = R' - n. R' = IApril By Theorem 20, We note also, e + e' = E. These equations give r = e= N. n = e- E - N - n' (R' - r') = = e - \1 + (e' - n') R - n'. Thus G is a dual of G'.