By Delio Mugnolo

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Let G = (V, E, g) be a simple graph. Assume that V is the disjoint union of node subsets V1 , . . , VN . , vertices in the same set Vi may not be adjacent), then G is called N -partite, and in particular bipartite if N = 2. If an N -partite graph is simple and if for all i each node in Vi is adjacent to any other node in Vj , j = i, then G is called complete N -partite. ,|VN | . 22. By definition a graph is bipartite if and only if it has a cut containing all edges. ) and such that each edge has endpoints coloured differently.

Morover, also property (ii) holds: if v ∈ V is not along W, then fn+1 (v, V) = 31 fn (v, V); on the other hand, if v is indeed along such a walk, then the flow along the incoming walk’s edge is decreased by and the flow along the outgoing walk’s edge is augmented by , whereas fn+1 ’s flows along further edges incident in v are the same of fn ’s, so that again fn+1 (v, V) = fn (v, V). One also sees that fn+1 is Z-valued by construction. By definition of a walk, there is only one (oriented) edge e with initial endpoint vso and such that fn (e ) = fn+1 (e ) – in fact, fn (e ) > fn+1 (e ).

A cycle Cn is bipartite if and only if n is even. , blue). 24. Prove that the Petersen graph is not bipartite. 25 (J. K˝onig, 1916). Prove that a graph is bipartite if and only if it contains no cycle of odd length. 26. ,mp has 1≤i