Identify true statements:(I) Every complete graph is Hamiltonian(II) Every wheel graph is Hamiltonian(III) Every complete bipartite graph is Hamiltoniana)I onlyb)I and II onlyc)II onlyd)I, II and IIICorrect answer is option 'C'. Can you explain this answer?

Question

Answer

True Statements:

(II) Every wheel graph is Hamiltonian:
A wheel graph is formed by connecting a single central vertex to all the vertices of a cycle graph. In a wheel graph, there is a Hamiltonian cycle that visits every vertex exactly once. The cycle can start from any vertex in the outer cycle and then move to the central vertex before returning to the starting vertex. Therefore, every wheel graph is Hamiltonian.

False Statements:

(I) Every complete graph is Hamiltonian:
A complete graph is a graph in which every pair of distinct vertices is connected by an edge. While it is true that a complete graph with three or more vertices is Hamiltonian, this statement does not hold for complete graphs with one or two vertices. In a complete graph with one vertex, there is no Hamiltonian cycle as there are no other vertices to visit. In a complete graph with two vertices, there is only one edge and no Hamiltonian cycle can be formed. Therefore, not every complete graph is Hamiltonian.

(III) Every complete bipartite graph is Hamiltonian:
A complete bipartite graph is a graph in which the vertices can be partitioned into two sets such that every vertex in one set is connected to every vertex in the other set. In a complete bipartite graph, there is no Hamiltonian cycle. This can be proven by considering the number of vertices and edges in the graph. The number of vertices in a complete bipartite graph is the product of the sizes of the two sets, which can be large. On the other hand, the number of edges is the product of the sizes of the two sets, which can be much smaller. Therefore, the number of edges is significantly less than the number of vertices, making it impossible to form a Hamiltonian cycle. Therefore, not every complete bipartite graph is Hamiltonian.

Conclusion:
The only true statement is that every wheel graph is Hamiltonian. The other two statements are false, as not every complete graph or complete bipartite graph is Hamiltonian.

Answer

All complete graph with more than two vertices are Hamiltonian.
All wheel graphs are Hamiltonian.
K_m,n is Hamiltonian only when m = n.

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