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.

Discrete 47.3

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

discrete 47.13

13. Let G be a connected graph with (see attachment). Prove that G contains exactly one cycle. Please send as word attachment

Graph Colouring 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)

Graph Colouring Problem

Please use words to describe the solution process: Let G be a graph with n vertices that is not a complete graph. Prove that x (G) < n HINT: If G does not contain k3 as a subgraph, then every face must have degree at least 4. *(Please see attachment for proper symbols)