On Hamiltonian-Connected and Mycielski graphs
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A graph G is Hamiltonian-connected if there exists a Hamiltonian path between any two vertices of G. It is known that if G is 2-connected then the graph G^2 is Hamiltonian-connected. In this paper we prove that the square of every self-complementary graph of order grater than 4 is Hamiltonian-connected. If G is a k-critical graph, then we prove that the Mycielski graph μ(G) is (k+1)-critical graph. Jarnicki et al.[7] proved that for every Hamiltonian graph of odd order, the Mycielski graph μ(G) of G is Hamiltonian-connected. They also pose a conjecture that if G is Hamiltonian-connected and not K_2 then μ(G) is Hamiltonian-connected. In this paper we also prove this conjecture.
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