have odd degree. Furthermore, each Eulerian trail ofGbegins at one of these odd vertices and ends at the other.Pen stroke problem.How many pen strokes is needed in order to trace a givendiagram such that each segment of the diagram is traced exactly once?3. Theorem 6.5 (Theorem 3.13). IfGis a Hamiltonian graph, then for every nonemptyproper subsetSofV(G),ω(G\S)≤ |S|,whereω(G) denotes the number of components in a graphG.Question: We have seen that the graphGof Figure 6.10 is not Hamiltonian. Showthatω(G\S)≤ |S|for every nonempty proper subsetSofV(G). What does this sayabout Theorem 6.5?4. Theorem 6.6 (Theorem 3.7). LetGbe a graph of ordern≥3. If deg(u) + deg(v)≥nfor each pairu,vof nonadjacent vertices ofG, thenGis Hamiltonian.5. Corollary 6.7 (Corollary 3.8). IfGis a graph of ordern≥3 such that deg(v)≥n/2for each vertexvofG, thenGis Hamiltonian.6. (Corollary 3.9). LetGbe a graph of ordern≥2. If deg(u) + deg(v)≥n-1 for eachpairu,vof nonadjacent vertices ofG, thenGcontains a Hamiltonian path.7. Theorem 6.8 (Theorem 3.10).Letuandvbe nonadjacent vertices in a graphGofordernsuch that deg(u) + deg(v)≥n. ThenG+uvis Hamiltonian if and only ifGis Hamiltonian.