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A few small details need to be clarified here. First, a graph containing only one vertex qualifies as bipartite: It is not necessary for both colors to actually appear. Equivalently, one set, X or Y, can be empty. Also, any graph that has no edges is bipartite. This can be seen more easily if we reword the definition of a bipartite graph to say that no edge has both of its endpoints in the same set, X or Y. In terms of coloring, no edge has two endpoints 25 26 Graph Theory of the same color. Clearly a graph with no edges satisfies this condition no matter how the vertices are colored.

There is a systematic procedure for solving problems like D24 (see problem D45), and it leads to the conclusion that there is a one-to-one correspondence between the spanning trees in K5 and all possible 3-letter codewords using letters from the set fA; B; C; D; Eg. There are 53 , or 125, such codewords, and therefore an equal number of spanning trees in K5 . More generally, in Kn , each spanning tree corresponds to a short codeword containing n 2 symbols chosen from the set of n vertices of Kn .

If so, find one. If not, explain why. (a) (6, 6, 4, 4, 4, 4, 4, 4, 2, 2, 2) (b) (4, 4, 4, 4, 4, 3, 3, 2, 2) (c) (5, 5, 5, 5, 5, 5, 4, 4, 4) (d) (5, 4, 4, 3, 3, 2, 2, 2, 1) (e) (5, 5, 4, 3, 3, 3, 3, 3, 3) (f) (5, 5, 5, 5, 4, 4, 4, 3, 3) C27 Suppose that G is a bipartite graph with a particular division of the vertices into two sets X and Y, as in the definition of a bipartite graph. Make up a reasonable definition for the bipartite complement of G, another bipartite graph with the same vertices.

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