By G. Duvaut, J. L. Lions

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The cut space is the orthogonal complement of the cycle space and so C is an edge cut of G if and only if f(C ) is an edge cut of G ′ . Further, if C 1 ∩ C 2 = ∅, C 1 is an edge cut and C 2 is a cycle of G , then f(C 1 ) ∩ f(C 2 ) = ∅, f(C 1 ) is an edge cut and f(C 2 ) is a cycle of G ′ . Let v be a vertex of G . Let N (v) denote the set of edges incident to v. t. inclusion) edge cut. 7). Further it is not difficult to see that in a 2-connected graph, any minimal edge cut with the above property must be the neighborhood N (u) for some vertex u.

We can color edges as well as vertices. A proper edge-coloring is a coloring where the edges incident to the same vertex get different colors. The edge-chromatic number χ ′ (G ) is the minimum number of colors in a proper edge-coloring. 3. A graph of maximum degree ∆ has edge-chromatic number ∆ or ∆ + 1. A proof may be found in [BB]. Let us mention that it is algorithmically hard to say, for a given graph, where the truth is. 38 CHAPTER 2. 8 Random graphs and Ramsey theory The G (n,p) model of a random graph is perhaps the best known random graph model.

Then G ′ is a bipartite graph with parts V1 ∪ (U 2 \ {u}) and U 1 ∪ (V2 \ {v}). It is easy to verify that these twists preserve perfect matchings. Are twists sufficient for a description of the matching preserving bijections between bipartite graphs? The answer is no, one more operation is needed: Let G 1 ,G 2 ,G 3 be bipartite graphs with bipartitions (V1i ,V2i ), i = 1,2,3, and having pairwise disjoint vertex sets. We further assume that |V1i | = |V2i | + 1. Let ai ,bi be vertices from the same part V1i of G i , i = 1,2,3.