Eulerian and Non-Eulerian Graphs

Let G be a connected graph that is not Eulerian. Prove that it is possible to add a single vertex to G together with some edges from this new vertex to some old vertices so that the new graph is Eulerian. Please see attachment for background and hints.

Trees: Vertex; Cycle

Let G be a graph in which every vertex has degree 2. Is G necessarily a cycle? *Please see attachment for additional information. Thanks. Use words to describe solution process. Use math symbol editor like LateX, please no stuff like <=.

Discrete 47.3

3. Let d1,d2...dn be .... prove that d1...dn are degrees of the vertices. (see attachment for full question)

Graphs : Connectedness and Cycles

13. Let G be a connected graph with (please see the attachment). Prove that G contains exactly one cycle.

Graph Coloring Problem

Please use words to describe the solution process. Let G and H be the graphs in the following figure (see attachment): Please find x(G) and x(H).