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

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

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, ...

(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.

Define homogeneous recurrence relation. Write the first order and second order homogeneous recurrence relations with constant coefficients giving an example for each. Solve the following recurrence relation:  for  given that 

(a) Solve  for  by Substitution method.

(b) Solve the recurrence by using iterative approach :

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.