## Abstract

Two Hamiltonian cycles \(C_1=\langle u_0,u_1,u_2,...,u_{n-1},u_0 \rangle \) and \(C_2=\langle v_0,v_1,v_2,...,v_{n-1},v_0 \rangle \) of a graph *G* are *independent* starting at \(u_0\) if \(u_0=v_0, u_i\ne v_i\) for all \(1\le i\le n-1\). A set of Hamiltonian cycles *C* of *G* are *k-mutually independent* starting at vertex *u* if any two different Hamiltonian cycles of *C* are independent starting at *u* and \(|C| = k\). The *mutually independent Hamiltonianicity* of graph *G* is the maximum integer *k*, such that for any vertex *u* of *G* there exist *k*-mutually independent Hamiltonian cycles starting at *u*, denoted by IHC(\(G)=k\). The *Cartesian product* of graphs *G* and *H*, written by \(G \times H\), is the graph with vertex set \(V(G) \times V(H)\) specified by putting (*u*, *v*) adjacent to \((u', v')\) if and only if \((1)\;u = u'\) and \(vv' \in E(H),\) or \((2)\;v = v'\) and \(uu' \in E(G)\). In this paper, for \(G = G_1 \times G_2\), where \(G_1\) and \(G_2\) are Hamiltonian graphs, IHC(\(G_1 \times G_2) \ge \) IHC(\(G_1)\) or IHC(\(G_1)\) + 2 is proved when given some different conditions.

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## Acknowledgments

The authors would like to thank the anonymous reviewers for their helpful comments. This research was supported in part by the Ministry of Science and Technology, Taiwan, under Grant MOST 104-2221-E-260-005.

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Wu, K., Wang, Y. & Juan, J.S. Mutually independent Hamiltonianicity of Cartesian product graphs.
*J Supercomput* **73, **837–865 (2017). https://doi.org/10.1007/s11227-016-1804-x

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### Keywords

- Hamiltonian
- Mutually independent
- Hamiltonianicity
- Cartesian product