Download e-book for iPad: An account of some aspects of combinatorial mathematics by L. Mirsky

By L. Mirsky

ISBN-10: 0124985505

ISBN-13: 9780124985506

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The graph G is called finite or infinite according as N is finite or infinite. If x,y E N, x # y and e = {x,y } E E, then the nodes x and y are said to be linked by the edge e. We also say that e and x,and equally e and y , are incident. Two edges which are incident with the same node are said to be concurrent. Jt is often helpful to picture a graph as a set of points (nodes) in which certain pairs of points are linked by lines (edges). A class of graphs which is particularly important in the discussion of many problems of transversal theory is the class of bipartite graphs.

I but of the elements of the index set I specifying them. The two sets of objects E and I are then linked by an 'incit At this stage of the discussion, the notion of symmetry is merely intuitive: its precise meaning will emerge later. 3 33 DUALITY dence relation’. Thus, if eE E, i E I, the statement e c A i may o r may not be valid. We have therefore two sets E and I together with certain ordered pairs ( e , i ) , namely those for which eE Ai. I n other words, we have E, I, and a subset of the Cartesian product E x 1.

I was comparatively difficult. Other proofs of this or of closely related results have since been given by a number of writers, among them W. Maak ( l ) , Marshall Hall Jr. (2), Weyl ( l ) , Everett & Whaples (I), Halmos Vaughan (l), D. Gale (2, 143-6), and R. Rado (11). The first of the proofs of Hall’s theorem offered above is that of Halmos & Vaughan. The very transparent argument can be adapted for coping with more general situations (cf. for example Exs. 3). 2). 2). D. J. A. Welsh (7)exploited the method of reduction to obtain generalizations of Hall’s theorem.

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An account of some aspects of combinatorial mathematics by L. Mirsky


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