Chapter 11.1, Problem 12E

a)

To determine

To prove: Each odd vertex of $\mathcal{G}$ is the initial vertex of a path consisting entirely of new edges (in ${\mathcal{G}}^{\text{'}\text{'}}$) and leading to another odd vertex.

b)

To determine

To prove: By improving the result of (a) that the odd vertex of $\mathcal{G}$ can, in fact, in fact, be partitioned into two equal sets, $\left\{v_1,v_2,\mathrm{....},v_m\right\}\text{ and }\left\{w_1,w_2,\mathrm{....},w_m\right\}$, with edge disjoint paths consisting entirely of new edges between $v_i\text{ and }{w}_i$.

c)

To determine

The conclusion that all the pseudographs obtainable from $\mathcal{G}$ by duplicating edges, the algorithm yields an Eulerian pseudo graph of minimum weight.