46.9 Discrete

9. Let G be a graph. Prove that....

46.11 Discrete

11. Let G be a graph with... (see attached)

48.1 Discrete

1. We noticed that a graph with more than two vertices of odd degree cannot have an Eulerian trail... (see attached).

48.2 Discrete

2. A domino is a 2x1 rectangular piece of wood. On each half of the domino is a number, denoted by dots. In the figure, we show all C(5,2) = 10 dominoes we can make where the numbers on the dominoes are all pairs of values chosen from {1,2,3,4,5} (we do not include dominoes where the two numbers are the same)... (Please s ...continues

48.3 Discrete

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.

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)