Is a Hamiltonian graph Eulerian ? Is a Eulerian graph Hamiltonian ? Show with the help of a suitable example.
In graph theory, Hamiltonian and Eulerian graphs are two distinct concepts, each with its own set of properties. A Hamiltonian graph is one that contains a Hamiltonian cycle, which is a cycle that visits each vertex exactly once. On the other hand, an Eulerian graph is one that contains an Eulerian circuit, which is a cycle that traverses each edge exactly once.
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To multiply two n-digit numbers, one must do normally
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
Verify, “If an undirected graph has exactly two vertices of odd degree there must be a path joining these two vertices.”
(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
Show the followings:
a) Show that for a subgraph H of a graph G, ∆(H) ≤ ∆ (G)
b) Show that
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