Prove the followings:
a) the sum of the degrees of the vertices of G is twice the number of edges
b) If W is a u-v walk joining two distinct vertices u and v, then there is a path joining u and v contained in the walk using the principles of mathematical induction
c) A connected graph G is Eulerian if and only if the degree of each of its vertices is even.
d) If G is a connected planar (p,q)-graph, then the number r of the regions of G is given by r = q - p +2

Is a Hamiltonian graph Eulerian ? Is a Eulerian graph Hamiltonian ? Show with the help of a suitable example.

(a) Find chromatic number of bipartite graph Km, n. 
(b) Is every subgraph of a regular graph regular ? Justify. 
(c) Construct a 5-regular graph on 10 vertices.

Verify, “If an undirected graph has exactly two vertices of odd degree there must be a path joining these two vertices.”

Define a recurrence relation. Describe the following problems with the help of examples which can be solved through Divide and Conquer technique and Show its recurrence relation.
(i) Binary Search
(ii) Merge Sort
Solve these recurrence relations with a substitution method

a) Find a recurrence relation and initial conditions for 4,14,44,134, 404, …
b) Find the generating function of 2, 4, 8, 16, 32, ...

Prove the followings:
a) the sum of the degrees of the vertices of G is twice the number of edges
b) If W is a u-v walk joining two distinct vertices u and v, then there is a path joining u and v contained in the walk using the principles of mathematical induction
c) A connected graph G is Eulerian if and only if the degree of each of its vertices is even.
d) If G is a connected planar (p,q)-graph, then the number r of the regions of G is given by r = q - p +2