Mutually Independent Hamiltonian Cycles of Cn x Cn

In a graph G, a cycle is Hamiltonian cycle if it contain all vertices of G. Two Hamiltonian cycles C_1 = 〈u_0, u_1, u_2, ..., u_{n−1}, u_0〉 and C_2 = 〈v_0, v_1, v_2, ..., v_{n−1}, v_0〉 in G are independent if u_0 = v_0, u_i = ̸ v_i for all 1 ≤ i ≤ n−1. In G, a set of Hamiltonian cycles C = {C_1, C_2, ..., C_k} is mutually independent if any two Hamiltonian cycles of C are independent. The mutually independent Hamiltonicity IHC(G), = k means there exist a maximum integer k such that there exists k-mutually independent Hamiltonian cycles start from any vertex of G. In this paper, we prove that IHC(C_n × C_n) = 4, for n ≥ 3.

• Kai-Siou Wu
• Justie Su-Tzu Juan

• Hamiltonian
• independent
• cycle
• Cartesian product
• mutually independent Hamiltonicity

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