3.1.10. Let M and N be matchings in a graph G, with |M| > |N|. Prove that there exist matchings M' and N' in G such that |M'| = |M| – 1, |N'| = |N|+ 1, and M', N' have the same union and intersection (as edge sets) as M, N. %3D

7. a) How many different paths of length 2 are there in the undirected graph G in Fig. 11.43? b) Let G = (V, E) be a loop-free undirected graph, where V= {U, v2. ..., v) and deg(v,) = d,, for all I sisn. How many different paths of length 2 are there in G? Figure 11.43

5. Let G = (V, E) be a bipartite graph with partite sets X and Y. Suppose X = {x₁,x₂,...,xn} and Y = {y₁, y2,...,Ym}. Show that Σdc(x₁) = |E|. i=1 m (HInt: Think graph fairy). (Also, note that Σdc(y;) = |E| as well, but just prove the above). j=1

3.2.21. Draw a connected path starting at (0,0) that, if continued, would pass through all of Z ×Z.