Is it true or false? If it is true, include a (short, but clear) argument why it is true, and if it is false, include a concrete graph which shows that the claim is false. a) If all vertices in a connected graph have even degree, then for whichever two vertices u and v in the graph you choose, there is an Eulerian trail between u and v. b) Given a graph G we construct a new graph H by adding a new vertex v and edges between v and every vertex of G. If G is Hamiltonian, then so is H. c) We know that if a graph has a walk between u and v it also has a path between u and v, for any two vertices u and v. Is it always true that if a graph has a circuit containing u and v it also has a cycle containing u and v? d) The complete bipartite graph K?,?(lowered indicies) is Hamiltonian if and only if m = n ≥ 2.
Is it true or false? If it is true, include a (short, but clear) argument why it is true, and if it is false, include a concrete graph which shows that the claim is false.
a) If all vertices in a connected graph have even degree, then for whichever two vertices u and v in the graph you choose, there is an Eulerian trail between u and v.
b) Given a graph G we construct a new graph H by adding a new vertex v and edges between v and every vertex of G. If G is Hamiltonian, then so is H.
c) We know that if a graph has a walk between u and v it also has a path between u and v, for any two vertices u and v. Is it always true that if a graph has a circuit containing u and v it also has a cycle containing u and v?
d) The complete bipartite graph K?,?(lowered indicies) is Hamiltonian if and only if m = n ≥ 2.
Trending now
This is a popular solution!
Step by step
Solved in 2 steps
