By Claude Berge

ISBN-10: 0444103996

ISBN-13: 9780444103994

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A inl l ) (a2II a,j z ... a2jn ... (a, fl 2) kl n, ... a, kn p). ,IN,I=n,. By proposition 1, the total number of such monomials is therefore n n! , np> = nl ! n , ! n,! ' If we put a? = a? = = a; = a, for all i, we obtain the desired formula. +. D. Proposition 3. , np Clearly, (a, + a, + + up)" = (a, + a2 + -.. + up)(a, + a, + + up)"-' , - a . , n p1 n a;1 a? = ... i( . , np a;l ... a;'- ... a: . D. We are now ready to consider the problem of counting the number of ways to choose a set E c P , ( X ) such that the simple graph ( X , E ) is a tree.

Show that a vector z E K” is a topological cycle if, and only if, the scalar product ( z, w = 0 for all topological cocycles w. Do the same for topological cocycles. > CHAPTER 3 Trees and Arborescences 1. Trees and cotrees A tree is defined to be a connected graph without cycles. A tree is a special kind of 1-graph. , a forest is a graph without cycles. Theorem 1. Let H = ( X , U ) be a graph of order I X I = n > 2. The following properties are equicalent (and each characterizes a tree): (1) H is connected and has no cycles (2) H has n - 1 arcs and has no cycles, ( 3 ) H is connected and contains exactly n - 1 arcs, (4) H has no cycles, and i f a n arc is added to H , exactIy one cycle is created, (5) H is connected, and if any arc is remooed, the remaining graph is not connected, (6) Every pair of rertices of H is Connected by one and only one chain.

If (X,W ) is a cotree, we shall s'how that (X,V ) is a tree by using the Arc Colouring Lemma (Chap. 2). V contains no cycles. Let c' E V. Colour arc black. Colour the arcs in V - { u } red, and colour the arcs in W green. Since ( X , W ) is a cotree, G has a cocycle of black and green arcs that contains arc c. Thus there cannot be a cycle of red and black arcs containing arc u. Since c was selec'ted arbitrarily, there cannot be any cycle in V. 21 If w E W, the set V u { IV } contains a cycle. Colour arc w black.

### Graphs and hypergraphs by Claude Berge

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