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# Mutually independent Hamiltonianicity of Cartesian product graphs

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## 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 (uv) 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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